,this is a collection of mathematical   
constants444  ,these numbers have been  
downloaded from3                        
http3_/_/www4cecm4sfu4ca_/projects_/,,is
c_/,id4html  ,an index of high precision
 tables of functions can be found at3   
http3_/_/www4cecm4sfu4ca_/projects_/,,is
c_/rindex4html  ,you can find           
information about some of the constants 
below at3                               
http3_/_/www4mathsof4com_/asolve_/consta
nt_/constant4html  ,thank you to ,simon 
,plouffe 7from ,simon ,fraser           
,university7 for his kind permission to 
distribute this collection of constants4
                                        
----------------------------------------
-------------------------------------   
,contents --------    #a-f_/7,pi#b7 to  
#ejjj digits4   #a_/log7#b7 the inverse 
of the natural logarithm of #b to #bjjj 
places4   #a_/sqrt7#b99,pi7 to #ajbd    
digits4   sum7#a_/b7bn71n#j44;infinity74
 to #ajbd digits4   #c_/7,pi99,pi7 to   
#bjjj digits4   arctan7#a_/b7 to #ajjj  

digits4   ,the ,artin's ,constant       
product7#a-a_/7p9999#b-p71pprime7   ,the
 ,backhouse constant   ,the ,berstein   
,constant   ,the ,catalan ,constant     
,the ,champernowne ,constant            
,copeland-,erdos constant   cos7#a7 to  
#aejjj digits4   ,the cube root of #c to
 #bjjj places4   #b99997a_/c7 to #bjjj  
places   ,zeta7#a1b7 ot the derivative  
of ,zeta function at #b4   ,the         
,dubois-,raymond constant   exp7#a_/;e7 
to #bjjj places4   ,gompertz 7#ahbe7    
constant   exp7#b7 to #ejjj digits4     
exp7,e7 to #bjjj places4                
exp7-#a79999;exp7-#a7 to #bjjj digits4  
,the exp7gamma7 to #ajbd places4        
exp7-exp7#a77 to #ajbd digits4          
exp7-gamma7 to #ejj digits4   exp7-#a7  
exp7,pi7 to #ejjj digits4               
exp7-,pi_/#b7 also i9999i to #bjjj      
digits4   exp7,pi_/#d7 to #bjjj digits4 
exp7,pi7-,pi to #bjjj digits4           
exp7,pi7_/,pi9999,e to #aajj places4    
,feigenbaum reduction parameter         
,feigenbaum bifurcation velocity        

constant   ,fransen-,robinson constant4 
gamma or ,euler constant                
,,gamma7#a_/c7 to #bef digits4          
,,gamma7#a_/d7 to #eab digits4   ,the   
,euler constant squared to #bjjj digits4
   ,,gamma7#b_/c7 to #bef places   gamma
 cubed4 to #ajbd digits4                
,,gamma7#c_/d7 to #bef places4          
gamma99997exp7#a7 to #ajbd digits4      
#b9999sqrt7#b7 a transcendental number  
to #bjjj digits4   ,si7,pi7 or the      
,gibbs ,constant to #ajbd places4   ,the
 ,gauss-,kuzmin-,wirsing constant4      
,the golden ratio 3 7#asqrt7#e77_/b to  
#bjjjj places4   ,the ,golomb constant4 
,grothendieck's majorant4   #a_/,w7#a71 
the inverse of the omega number 3       
,w7#a74   ,khinchin constant to #ajbd   
digits4   ,landau-,ramanujan constant   
,the ,lehmer constant to #ajjj digits4  
,lemniscate constant or ,gauss constant4
   ,the ,lengyel constant4   ,the ,levy 
constant4   log7#aj7 the natural        
logarithm of #aj to #bjjj digits4   ,the
 log#aj of #b to #bjjj digits4          

log7#b71 natural logarithm of #b to     
#bjjj places4   log7#b7 squared to #bjjj
 digits4   log7#b99,pi7 to #bjjj places4
   log7#c71 natural logarithm of #c to  
#bjjj places4   log7#d7_/log7#c7 to     
#ajbd places4   -log7gamma7 to #ajbd    
digits4   ,the log of the log of #b to  
#bjjj digits1 absolute value4   log7,pi7
 natural logarithm of ,pi to #bjjj      
places4   ,the ,madelung constant       
,minimal y of ,,gamma7x7                
,bessel,i7#a1b7_/;,bessel,i7#j1b72      
,the omega constant or ,w7#a74          
#a_/7one-ninth constant7   ,the ,parking
 or ,renyi constant4   ,pi_/#b99sqrt7#c7
 to #bjjj digits4   ,pi9999exp7#a7 to   
#bjjj digits4   ,pi#b to #ajjjj digits4 
,the ,smallest ,pisot-,vijayaraghavan   
number4   arctan7#a_/b7_/,pi1 to #ajbd  
digits4                                 
product7#aa_/n9999#c1n#a44;infinity7    
exp7,pi99sqrt7#afc771 the ,ramanujan    
number   ,the ,robbins constant   ,salem
 ,constant   sin7#a7 to #ajbd digits4   
#b99997a_/d7 to #ajbd places4           

sqrt7#c7_/b to #ejjj digits4            
sum7#a_/;binomial7#b99n1n71n#a44;infinit
y7 to #ajbd digits4                     
sum7#a_/7n99binomial7#b99n1n7711n#a44;in
finity72 to #ajbd digits4               
sum7#a_/nn1n#a44;infinity72 to #ajbd    
places4   ,the ,traveling ,salesman     
,constant   ,the ,tribonacci constant   
,the twin primes constant4   ,the ,varga
 constant1 the one_/ninth constant      
-,zeta7#a1a_/b74   -,zeta7-#a_/b7 to    
#bef digits4   ,zeta7#b7 or ,pi9999#b_/f
 to #ajjjj places4   ,zeta7#c7 or ,apery
 constant to #bjjj places4   ,zeta7#d7  
or ,pi9999#d_/ij to #ajjjj places4      
,zeta7#e71 the                          
sum7#a_/n9999#e1n#a44;infinity7 to #eab 
digits4   ,zeta7#g7 to #eab places 3    
sum7#a_/n9999#g1n#a44;infinity7         
,zeta7#i7 or                            
sum7#a_/n9999#i1n#a44;infinity7   ,the  
,hard hexagons ,entropy ,constant       
----------------------------------------
-------------------------------------   
#a-f_/7,pi#b7 to #ejjj digits4          

4#cibjgbhihadeigccgaccfgbcbbjgdafcdaffeg
chdgceaiffebjgjfibfcdehjhfcdifabgdbbfeh 
#gcebhebgddcefddfbfhigdcahbffeccdcjhhefh
bejiiaefbhcidjhcdhiebeehcighfiajaiajjdd 
#afhbhgdahhcgfcjcdfjijcddabhadbgbebcbbhj
jhahdfiejeddehhideajdcdibcchdeeaihfjbce 
#dghjacgebhgdhhhdebedfibccbfahahceggaajh
gghahedbebighhhdaghfhegfhfdfagbgehaaefa 
#ccjfcjdbdaibajciibcgahddjfcjjehajgbigia
cfghajbcbiaggchgbchhechfifddcahbfdecece 
#ijeijgfadddiafgbhhbaeiaghifgbafbfbhjebh
bgehifjfgjchadgfbgdbadchajbhgddbjbjiaad 
#bhcjcajhibhggiahbcehcahhgbjdeghcfjcggbd
iehgbgedjicgcbeigabdjbcejjficcidahhgjhh 
#febbgcahbgijhhfeheadbicaibfeeiahaihhigd
hffbddcdjhfbieacaehabjebhjibjdgejdgdhaf 
#dcjbdgjbcigcgahbaeaecdiaghfadhgecdiajjc
chafgcdfjghfhccbjhbiahahecjjfhiiijijgba 
#gebdbaddaecdijcjbidichdaifchajcdiabihed
hafbgedcbjfibfafhihhcdiijdbfgbgidefcbgi 
#biiejdahjgacajbjhhgfegehidibbegiddhddjg
cjfhiabeceggeccbfbgehjebiaabfeeegiebhae 
#cbejdjffcfbhfejcabiafijajaegggefaghbhai
fajejhgbgbahgebfchdjjgecifcidfijahibgcd 
#cifgaaaecbbehjcddeeccidaefhhehfcbddecja

fdigdbeaiafecafddacgafjigaaecabdejhibdc 
#bijedihbdfdiigeacdaehjddabhhahebgchfgbf
efeechahccjcjahadfcejgjibggaaifidcgbcde 
#fggehbfjhfdgafcdbedchhijdbgaejdajjbdaeg
gacgahhfjifehfbacacbgbdedbihijahjdgeaac 
#aecdaabfciidjcfiefibgdejijehedhcfaiebgg
ecghhjbjdhbhecdaahfbjijbffcchhibjhcgiig 
#ffjchfbaefhcdabcbcegadeebhajcdghhjidbif
dfiiegfcddjgbjeigifcghcicifiiibiabfhhei 
#bhjdidhfghcabjbhcifcbdjhbcadadgjbifebhd
ahacaacahchgcbcijeacfajahdebcjfdiaiacbh 
#cddbjdejfechbajdhhcchcfbiiidahgbejbddia
bijifhdfcjbdcdabcjicibfjicgbajfcggfcceg 
#ffhgafcbejdcecbedjgbjgefhbdjdciadifbfdg
gdihciaedhcgjcefafeabcjijcbfchedaegfbdf 
#eabcfcdbhgeigffbbeecidhaiddihcdibdcdcbf
ebgbjdagjfdefhaeacgfjeehajggafhdifadbcd 
#fbdbhdcbcgjafjabhegbjeeffjjghahjcgjbjgj
hcjbfibfbecfiggecciehacjdgbghcaeghieebg 
#jiifdhebdjeeffcegibjiejfifedjfchiadhgja
bjadaaafefdcbegdfbhfbedebdhiafbhbeceibd 
#bhcajiacehghhcabaggehdbeciifeiibfhjgcfd
jbbfacajjgaejdbajbfjcahhfcaecbffbfghbee 
#giccfhdfbfjhfejabgijbdfabijddhbdhicchde
chbeicjfbicbdjebhhjiiadgjheafcccgfddbei 

#jifidbdegihbhfifhahhddibgeabiaidebacjee
baibbegiabfhdbhfdfgcgdjdgdhgfbijhbgabbc 
#ihhjhjdfaicfgdehgjjbfihgjggifchccbeagdc
hjbdgicbgghcgfcaiicahcdaafefiecedfhhihf 
#ehggjijjffchihdgdjcdgeaicfgdjbgehdhiihi
iaffajjdjddcjgagfgeaaedjfcegdhiijbfdhdi 
#ihehfejigdhffhiieiijjjedfcfedhbghafhfei
gfijbjeebbjcddaaieeidfaijiehhcgaiifgeie 
#afcbhfbcchejfffiaccedageibjhdhabihafiej
bbdghebajfjbcjgagjbjjcbdjigibchaeedcijd 
#gfefbbdecgbaaffjibidajhcdggdgbfagcjbeei
ideajcicahbfhibaccdjbbfigehcjahebhacfgc 
#cacghgbhdbhgjddeafghfjjebcdccjehicbeecc
hfifahacfffbebfjbcachfhageihafjedefdhcj 
#hbcidacgfdjfcdfbcecicjeiaaeegjcgaehhhig
aebhhiifahibdhadajfaifdcieegjifjjajdghe 
#fgfejadgjjeciegccddjddibbfccajccbjhgeda
iegdfcggdjhbieheefahgjgciifgjeifibchacj 
#cbgefjjaehebhadjddfcdbaaihahhffgdgbcihf
ihhhigjbbhbecbggdbjijjfjhgcgjgigibcffca 
#jhgehdjfebagcdiafbfdgbacijifbhigefcjcea
abghefahjicghdiagahigeddagcahgiiggcedca 
#fheafdeebbacgbegbcedghhggffhiciiihjiafj
iaiifdgfgfagjijcddbjddfbfjddchicaiiggcg 
#gidiaediadiiejgfagcfgabcigfbdeddeffbhhd

chfigbajjjhibfhdibijajhaiceajgiddgaidad 
#hdbehabgbdhabdhchibabhbhdjichifcafdccfg
agihfccdbjbdgigggibhhgjahadehcbihhchieh 
#hcbibibfecahiidiadeabbbicjejcgicdagdcbc
afffhfbagjjaegjefffdhgdibcghafaijcgaecj 
#igjfgjjidcffhfciaeahehgheeijddgffecheji
jccefjhihefabehhicebiffiifjefeceebdahde 
#bihjhchheihhbjhfcjgibchcifeddcdicahigjb
afbdfcedefhjbbciedghbcbcciiiijeghjeecge 
#bchaffceigfjchjjfcjccbfhfbaebfdehffgegi
agfdaidbdichicjeagfbejigibbgeecaaahcgaj 
#gdeaabacefhfdhbiigicebehabgggdfjagffgjb
cgdgjabdfidihedchhhicdbeeghhdceghggiidh 
#chhgfdhdchafcfdcbcefahegjffeejdedgfhefd
afjdjjcgbafcfhhddcgajjjhfaigdfifcbdhgba 
#bhedejgccbbgfibgacahcbidcegccddajbaejfg
jfhfdchabbhicghbajcbaicahhidhifefhbjdff 
#hjiifgejfbjfcffhiffjcfchhgeifaffhiggbbg
dccaijibdbijjdagfhbjicefhgccbeaebcdjgia 
#iabhacjceadacbeefddjeahiggiiiabijbdbifc
fecjdjejbigacjcifieajiafjebcbacdfhjcbfc 
#fafcdgehbdgchiedhedbeiaefdbdffihjehgcje
edijijfjgfhdjagfbeccgebejdjiacaiidbcjce 
#chfjgibigjcifbjaiabhhehjjfibihcgceefbdi
dbigccaddiggbfjdijdbdjgbahahfbdjdebfhbf 

#hagjgaiddabbebgbchjhfedejicbgiahcgjfhdj
bhddgiecgieabigbheidcihaggaieddgcjgffeg 
#fhecdidiheicffagcdaahiddhhbgjdfdcaehdbj
bdhiceeabdeabcfbageejgfhaeegeceadghajie 
#igfdfhhjdgebjicjaiffbagigfbdffbdgcdggea
behhghbfcjfcecjicbdheeaibjdaihdafcegeei 
#ffheedfeibdjefcjbcidcbgbgiaeggjgdjdccfi
ajcedjbdiejeijbbibbdiihfahdecadbibjgcbj 
#ddafjchgcffeedbecficejjjffjeibbachcieag
facgdgecjicfbgjbadieedihcdfjccidhhebbag 
#iagejghgdcbahfeiddiehgdcechbfdgfibehacd
jdciaibcehiegfabhjdhbagebgghcajhffagbcj 
#cfhjbcdcjbdfcddgedbdcabegdgcedfebgdffbf
cgaajigjbjcjdjjidfgjijacgijabafcfibcdgi 
#ccdbfbachddejjbaadeeciffhefiagbfidfgcea
ahjjhidgjcddbefdedgdfggaffffbbcdbdefdib 
#agfdecgdjhghbecafafgdahbideiaajeiibadbf
gbdcgfifddfjgcbcbhegaagbgbfbagcjhebiebc 
#bfbhbedbfabfiajicgbcjbgjjeceddhcieabedf
hbidighhjaagbdfdfbbiiaacgbfgejdddheiccd 
#eehcdajdjbeacajgbdcdjhbehhaijfhhcfdihhd
gifhdjgebfiddhheibhhjihfieedfedjdfjfhhg 
#jehgaehiaabjiajigehifdhfagbehaajiechfej
ahcjibbgdhbjejiacicigbddfigdbccfhhebejh 
#aedgchcjdadcahcgceegjcbfjaaadiheebiifhb

hfgfiibejdadgejefedehcaihibiddcggcaeecf 
#ciaigagahddgjjjhccjidaajciaiajdiebjbbdg
jicjcbgdcahdijjcdddadjcidhjdiibigefjhcb 
#higijaajd                              
----------------------------------------
-------------------------------------   
#a_/log7#b7 the inverse of the natural  
logarithm of #b4 to #bjjj places4       
#a4ddbfiejdjhhhifcdjgceiibdfhajjahibacgd
bffdeiedaebiheicdaceddidjficaajibaiahaa 
#hejgihheebffbbhicejfcdddifiigeahcjifebe
ddbeeeicajafhgafhceifdbgbjffbaehbbcdgic 
#cfbgdecgcfihhdgahdicfcjgjachgffcecbjaee
cchidcahiafffdhcgfdcabhfaedbdjdgdghdbbb 
#hidigijdgiejiaecjceacchehhjedifhhfehicj
ifiifcfhjcfaajeaajgefcjhddadedbgbaehbhc 
#ddidahiaiccijhegggaegijjddagabhjbdfhdhc
dacgdebbfieahbcfijaabcijidjcddeiifhecii 
#jfaacdbagbbhhfbghjbiaehjajfcjjfaigfgfbd
defebfjeiiejgcgecbdjfbefeehaedgeichaghc 
#jebcigbeeajgbdhacjggaefbfgedehjgeghagac
cjaicegcjjfafhgfaicgcgbihbfgehigdaefbch 
#agihcefgajcddcdhigejfhjgjeeahjhhdhfefac
hfhcbiaggcbahbicdiacifhdcajeicdedjbbjbe 
#ahfcficdebfbfibaejieeigaiajjbbaifgibbdc

badccdbddidagijgadeeaahdiicheibabbafgec 
#fecaacjjggdfcbgfgbjfdfabccgdaajhbaaiacg
iddcccihdhjegicajiabhggfjifgjbjjcgegehi 
#ihaehheahjfabfghhjiigfjiefbebejghdajbdh
dgjefijjgfhgfhjehdfacbghfedgdghbjbghjhf 
#eidfbjfjiajgdijaecbdhaiifigcjegijaebgbc
bdghgbihgdjihabedajjjccddhfhgegchbbcfdg 
#afdideddgecgjfgafgeiehiidbhjiihahbfghcd
ijacaffffccecdhjcfghihfiddfhhgjiaafffjd 
#igcecgbibehfjgbabidhfigcededjgjhjihcjfg
dhichcdabcgahfcadjjhceigifahhfeigehfhgd 
#ebeccjedfhibabigffdaegjdbjfbabeibdfcacf
ibdbafhjeijhggdjhccehacibhfffedaehdigaa 
#fbehgjfieefeghehhgdgfiifcabifiebejjdeic
gbfbgchijbfhjefficeeabhgbidcchcgbaiacaa 
#affejhhajjaehghfbfeeiaefcgiedjeeijefggh
bbcfhadjjcjifhhdcicdhjhfbbhdhahdgiacdef 
#ccadaaicjbchdjbfdjigbgdhdcfddifbaieddib
bddfebbbjdgagfcehfjgdgifeheefffjecdjihb 
#hfjihegdjbghhcgdccabfhhefcceddcdcjfighg
jahifdcehbfaciaahajjbebeiijbjgffahddcbi 
#hdhhcahdgbciaeiabgjacijdegjdggcegfdhcaj
ajbaaibhbigjhecbhifjicafhjceciaifdihfie 
#gcbfdcicgiadijcjhdhedgjfafdccghihefcdhb
chijjjjdefdbfaheefbbdificjiacifjcabebjb 

#bcgfgcgfjgdaechfbaafbdeeeaafejhfdcfgiia
bichicgabbeegbgebheecehejechhfbgedfibha 
#fgeejdjgcjciahihdchifdajebjcihiijbajghi
jggdajgdfgjgaedhgahgddeibghbfdhjcbegdec 
#bidjfhcfeebeddajcdfegcgcbjcaeachbbeabic
fadcgfbdadbbjbbejgadcgjcfigcjgcdfjidadh 
#ejajhfjcahicbcfjdaaccaaaaegddicggjbdiad
fhhadeecfjigbbhfafgbdbebgbjhhhhijfaeagd 
#eajebecaeeiaghcafbdgjbidcjaghjieicdbebc
gaigeabefabc                            
----------------------------------------
-------------------------------------   
#a_/sqrt7#b99,pi7 to #ajbd digits4      
4#cihidbbhjdjadcbfggiciidfjeiicdchahfhdg
ehehfcaafdicdfegffeibehbifgjfegibehiicj 
#ahchejabebcccijgcjficfdcjcjbeehhfbfceah
bfheeajiiaiedeeehcgbdbiifbabgcjfbeejggj 
#fcdebgjehbgbjdiicagefdeafcdehjgecjeigbe
cfdbgcbjhcffieicdghbgagjbiiiahfdaijfcde 
#fjcbhjhiccchhfjfgjdfecfebgifgafhficdaie
dggaaggbabjfecbecgecfiaccdghgejefjdbdje 
#egjdhhdbehahjdhbcagijcggbhjdiigagfccheg
ecfciibhciadjcahficbhcfidggagedhehbcigg 
#ejedddgibggfaaeejgjdabgjcififgbdhejdgcc
gfjchadhacibcijacjjefdfgfjbccefcjeegjjh 

#egjjgbffdaajjjaegbaefciecegghbcabcdajie
bfjijfibfijhibddefgbdeeedfgbajegdcibhia 
#ebefgceajicjchejfhjghcahceaihjfeeaifdfh
gdchahiihjafeieighahhggbadehhfafagdeiij 
#jejagabifjidjcffcacbichdfbjahfejdecjiif
fhadcafdiadcbdbajfjdagdeebidecibhbbaifh 
#hgiigibgahajfabedacgjafddecfcfgfebhgdfd
hdjfabbeiggdjcjbgegfcbjacgjidbbaideaagb 
#edfedgjgehddbadadbbejbhchjfahfheidacebe
geedggdedihjaecjeghcdiadgfacjbbjjgdbbhi 
#bjbghbajiccjbfccbgfehbgdbidcdacfabfdcch
dihjjegifcehghiddcgbgeagaaeejacedeheihh 
#icicgdeeahhidcdjgchcbjdiaeaihbifaicjgjg
agfagejhjccbijhfedgcfdbhbbfiaideijfgecg 
#iihhagabich                            
----------------------------------------
-------------------------------------   
sum7#a_/b7bn71n#j44;infinity74 to #ajbd 
digits4                                 
#j4hafdbaejijbahicadcgjhjgigcgecjebebbag
jccaacgeibjeebhjdcdabajijch             
#dcjeefadaideeecjjjfjdhecacbdhcigbfefage
ehhdcedhbjgiccicbdiccdbec               
#achejbcgjcdgjafheiahjcafbejafdacghhaiej
eecigbaacfbacgjaibcbhdebc               

#dbhcabcdaajcjaeggdffahgfihejffefjijhgge
ieggcefjhheibgjhbeefgjifa               
#aeaafjcbeehcfajadecdabgbhjiebbecjbffjdh
fafdhbieibjhebdggdigbedai               
#aaiabgaejjecchcdjgcfgdeacaggdeddaffiidh
jbaeecjigbfhdcijfafigbaje               
#iiehjfejcicgibigehgjjebgjdgafajjbhbigdb
hiiegcdfddejegjagjaajcjhb               
#ficjebihifbgffgcidjjbjiigciaaecijbeaafi
baaeficccahefdcfaicbhahhf               
#gcefbeicceebjichabgjaffbfedafdecigcgahj
abbgidiibadgijiiababeaehi               
#ggaibebiegfbahfiiidebbaichdcgdhgcfbhiea
aeiiefjhggfbcbedbdbajighh               
#hjcabdiehbccghdchjdccbhhjbdjdhgdfgjifef
feeejdiiebghhgfgahjceabee               
#cddcghdhbfifjjadjahaefiabfhcijajjfabeee
ihdfjcaaefdcaabhhjaiiedff               
#ghdiffjbadhgibcaecejhifdjjihbaifhijadhi
ehjcbafhedfedfajihghhdcji               
#ccgeadgecgabcfghbefgjefageeddijjfiffgic
gchiideaajedcjiidaajddifh               
#hegbbgabihhaajegahegbjhcehcafjiagdhhefe
hjgddbcabciefdeehegdjcgch               
#hdijddjccaajhjgdjffhahjahecdbjeajibddjc

eidjhbeicgfcbidbgfbciecbe               
----------------------------------------
-------------------------------------   
#c_/7,pi99,pi7 to #bjjj digits4         
4#cjcifceejibgjaccadccafchchifbiahbiafga
cjgfcbdjafgcifdfecfhbgjiefhbeaicfbhhfgj 
#fcbcegcfbghbaggfhfeeabhdjhffgccbhdeegae
hgdejdbahehjbiehbeebcgbjhjajfeddijddigg 
#iaehefbijehaahdhbfiedhbgiceibhfcgchchei
ieijgfebdgbggjeebgddghbechcjggbdjjfihhb 
#bfjiicabcefbeeeggcgbfechccfijijhbaaddde
fajijgbhgceajeegiajfegaaefgfiacfbjidbai 
#ccdfhdghgijcidhjjchadjggifhdigjidfceajd
cafjidhhcedaacjfchjegcjfeagghdjhfggcbcb 
#jdgjdfaibggedafceehibjdajeafciahfheigce
hfbjeaiffdhjibfahfbhibhjidhefbghihieddb 
#hehdhddeecefajdjhhbjhdjefciggajhaihaacg
ebjfcfbbiecaccgjadcgihhbdifeccjbijefdee 
#fgchfcdjhfjdeefgjgdbhecdjcfgbjdjijjeeab
effhgghbiefhebdcdbjicigceiecigfbdgfbeia 
#ghdhgfdhhjacadjhibdbcbedajfibefbcbeddih
cjiafcbfifjfehcciehedjijgcdifeejjdedfci 
#abcghibgibcbedhdhebecjgijahjidhbedcejgb
eiahfbbhcifecfiaeejehbejdghffcfjbgahcfj 
#cejbdgijifdcddhieefagabjebechgajbgeggif

cdfeedcgcbaabcccfhfbjigceddcfgbbajbceib 
#ccgdgiffhahefgdhdcedaedidibaaabaiajheij
aidgdefcfcijfbcfhjgiifbcjahjbfedijecfcb 
#hjafdddbcchgjihbggbccjbibaeegjfhcgggcdi
ageabhgdjdagcdaggicadaieaddbcdcggdeecgh 
#cedgbejhgfgejabdcbibjiggiceeijgcfcjffcf
gagbcjijhcdhdijibfhbdfdecfaddjaebhachbg 
#afabjhfiefgfdahbhgbhjeedghfdbdgidihgiba
adcadjefieagjfhicdccfcggbhejedijigfbddc 
#dbcbidcfhjjbihaebaecfbgdedgjgbehaijbcfa
bcajeihigehegcbidjfhiedhffhcjeeciehajja 
#afihjfhibaehbichchbadbgbceidhbfjeiebhea
gfejbaahbgifcigjajahajhjcjaejjcedcfeegj 
#ceigebeffjhdcihehjahcgiehhdbibfdheagceg
ijicddcdjhjfcchjdgdcaidijggchdfgedjdcce 
#hbghiggdfgcjhidgeehcjhahejjbijfcgdhgged
cedeaegfhdhghbichdecjcfiecacidfhaaahcba 
#afefdahcgdghbccgbifcifbaehgighjdbeahfgf
abejhjdbbehadhbaiagdchdedhcfhjgbibaahgf 
#gdchahbhefbjaafhhgbcjbeijbgejhbecghbhcf
gcfcigiadfggaeibdcaaigbjidfadaegeaibhhb 
#fhgheghchadiaiicegacigbafiifjijihadhifd
ehdhfecfhgcaeaabccjbjicdgfcfjhdbajebbcf 
#dejagegcgigbafhbajciebdfeagbifhjedbefdi
ciibiddagaghcgabfhefhgbgcgeedahehgcbjcg 

#hehddedcbjfjehdciaabjghgcjjagjjcfeifcag
ihhficddifdbdghidhfihdjefhdbccffhffjhgb 
#ajccaegfhfiefgdicfjdhgficedggehgeeccjgg
cjhgjcdfheccgigceeiejdbfdegdahccaagghgj 
#deaebhggajjhefeaeijeggecfbdcedjbgcicdgb
cijchgajdcfe                            
----------------------------------------
-------------------------------------   
arctan7#a_/b7 to #ajjj digits4          
#j4dfcfdgfjijjjhjfaafbadbefbcadfabaddjbj
bhecgjedbhfabjbfchajiccjhhgbjaighfdafegd
                                        
#agjecjjfjjbhcihdhhghibeefebihebbeaaijhc
geacejehahahafbejaaaeedgaecjefiiddajefb 
#jgaiccfbffafdhhjajaecbejbgeeihgibehjeea
fhechhiafgdghbcgbhfechgiciahjabeagaiidh 
#djacieehchaheaaejiejbafcccjfdichgbaedfj
igcbjgheeeeegbjhfjadfcbbgefebdbfgcjebah 
#jdegdfdjjhfigdejehchigcfchifdhijjbfdhfh
gghecghjabhbcfccabagafdeghadfhcfijjiicc 
#djebhhhbdhfbddefbchhaaijijaehidigfgiiga
igjaadifggfjjafdejjfbecjafhababefjiccec 
#jdacdiciffcjabicaibdbgdhdjbicafaaaidibj
fafbjhddaeicgbcfabgcaffhgfihafhgjbgeica 
#hieajcccigccbeibijcheabhibedeideibbdfcb

aefjighcfchjjiecgdiicbjidhfjgccidiahfdc 
#beafjbgdhbgicjdejcgccaggbeedfejdiifjhfg
eggjfbbgeddafbhejbbbgcgbcgaaigddgccffig 
#gcaheajfidjachaabfiiegggibefbgdhbeffjji
fbaafgbfgdhaaebgbhbgbbebjgbbeigbfhdbaeg 
#ajaiehggefbjiagjaeeggfhgjihffedbffhijcd
diceahjedgbhijjecgjghchabdbabhedgidcjcj 
#bdcfghdebfdffiicgfhchjhhggaijdabgfgcaae
icgdhjfafbhhccjcbjbhhjddfebciehifahibda 
#cjeaebgjhgfgbfdcidjjjgjddcibcedbddbefia
bbfigggaaeahibggagbbfddfcd              
----------------------------------------
-------------------------------------   
,the ,artin's ,constant4                
product7#a-a_/7p9999#b-p71pprime7       
,reference 3 ,wrench1 ,john ,w41 ,jr4   
,evaluation of ,artin's constant and the
 twin-prime constant4 7,english7 ,math4 
,comp4 #ae #aifa #cif--cih4             
#j4cgcieehacfaibjbbhhjedgbhjedcdfeafdaea
aafbibdi                                
----------------------------------------
-------------------------------------   
,the ,backhouse constant calculated by  
,philippe ,flajolet ,,inria ,paris to   

#acjj places4                           
#a4defjgdidhehbfhifgaciieieceaaafedceegf
ecaghcgdhdgacaedjbgjgjbd                
#cgdadjjaejfbfechihieeiifdecaidjahfjcjia
jiibeadcfaifcdgacedhfjgg                
#eafdiacababcadbibjceaggjabhcagdjecfiebg
diihhjbedhfibcjgjehjhebh                
#deaabdjecjjjagibighefajfgdiaigjhejjegge
jjedchgfiahjjfhhjcbaeihj                
#fbjbgcfcdagcefjdhafhbcbdcijigaicgiabhig
heejjijdaahbjjfhhicgdagj                
#ebdfjeebcajcifhabcdaegfebeeabdccabibggb
aeghehfcbjjedfieficaehac                
#bdfejjjdjijbcgjfffffgaagedgaebbcfefdjdd
ceacihaficchigcicjcicgjh                
#deehcjhcffcfgciedbjdfiighaebiicgdgibfbe
bbejiagffifefefcbagbffeh                
#ecaaahbfbgjfjgdedebajgbhfddgehfddbcagag
iaaeigebgfigiffaieajjecb                
#ejffgicgjcfagdicfdigcjifceaafjhhgadeija
bjacdjiahfidiiiigbieabjj                
#caifheefeghgieggaeddfjgbjagdcfgicacbjai
bggjhdfjhadbehicbgagageb                
#adjcejffidgabeehbfeeabecaceedeeabfbaeii
agedcbdiagfhgjdibgjcajff                

#hbdieeagaieiggcfjdddgdhhecjebafidbjebfd
hachbghgbfgiaehbfgiefhaf                
#ifbjdbifjahciahhdaegfdecfdibeafjjdhibdj
jaaaijbbdefghdebjbacahdd                
#fjgibbhjdjffggajbjidfdiijjcicgfigibdbic
egijgfjfgiadieaeiibiddcg                
#ijfbadjcjhhdadcfhegfdhijidibceajiiedcgh
bebfeaihcfhdhdhefijajaag                
#dfchiiahdeiaebgjciggdjdffgfgfgbhigaaeea
jacbgacbagdedfddcgejccdf                
#eiejjebbgjdadaeiedfjjhhfjgbecfbeeaadebj
ajiaaebgggbdjiideebiffac                
#fiiecahejiihgdiggdbjbahecdcbeeggacacaba
dbccegibgahchaeiiafhagej                
#fbeagfaiifadjieeghiiedjbebicjidiafbggdg
cbfgjafiihjgbhfdahiffgeb                
#higidigdfdejhiffcifcgcighfihafaccfahadh
ge2                                     
----------------------------------------
-------------------------------------   
,the ,berstein ,constant4               
#j4bhjafidiijbchfiaccjcfdcfdiabcjfgbjjjj
dbdhbacihabcf                           
----------------------------------------
-------------------------------------   

,the ,catalan ,constant4  ,as calculated
 by ,greg ,fee using ,maple ,release #c 
standard ,catalan evaluation4 ,this     
implementation uses #a bit_/term series 
of ,ramanujan4 ,calculated on ,april #be
 #aiif in approx4 #aj hours of ,,cpu on 
a ,,sgi ,r#djjj machine4   ,to do the   
same on your machine just type this4    
catalan 3 evalf7,catalan1#ejajj73       
bytes used#cgefighbgdh1 alloc#ecgbifh1  
time#chjgh4ie  here are the #ejjjj      
digits 7#ajjj lines of #ej digits each74
   it comes from formula #cd4a of page  
#bic of ,ramanujan ,notebooks1part ,i1  
the series used is by putting x-- -#a_/b
 4  in other words the formula used is 3
 the ordinary formula for ,catalan      
sum77-#a799997n#a7_/7b99n#a79999b1n#j44;
infinity7 and then you apply the ,euler 
,transform to it 3 ref 3 ,abramowitz    
,stegun page 1 page #af4  the article of
 ,greg ,fee that took those formulas    
appear in ,computation of ,catalan's    
constant using ,ramanujan's ,formula1 by
 ,greg ,fee1 ,,acm #aiij1 ,proceedings  

of the ,,isaac conference1 #aiij        
7,,maybe #aihi71 held in ,tokyo4        
catalan 3                               
#j4                                     
#iaeifeeidaggbaijaejedfjceadicbchdaajggd
adicgdbhafg                             
#bacdbffdihaaifbagfcjaiggfbedgfidgicefea
bibfaaeajfb                             
#dhegddbbfaiaifaiiegijcehihhjccbeheijeid
caeidgcgdha                             
#aehdjfiieccbjbhggccaidfjeaijchgbgdghafd
jhghfeijijb                             
#dgjfdhdaebafcjjjbbhgbgfdjidbchhbeiieggd
aejhhafcigd                             
#gjbebdhbjaaefjgjgfddhhchjghgccgjdhiijjh
fdggeaacbbe                             
#iigacdcdjgdhedjgeecbcjgfhefeccegfhjiehc
ebfjbaichbc                             
#bciejhjjgbjfhjceegfajdhbcegccidbcaiadih
bihcfahiigg                             
#jfijcfdjdahjhfbagidaajaiagecbgdcadiighb
ccigfajeeab                             
#bdggiecjcbdhgecgahghffehbhjhbcfjegjbbee
idaidhahjig                             
#ecejigaacaegabfaehjdbdbgbcfcfdcihejjagc

hbhgeiggigf                             
#ecjfhcgjjibihjhgchhgdiefajhicfeiggaidji
fhgbfhdddda                             
#ffhjdfbafbdccihfdhchiafbhjddhbhaejfbgcj
bbgdbjgchhd                             
#caagbbahbgbaijdgbbeehgjecaijhfhegcedbcd
ihecidihcji                             
#iaiaaeifgchhdfdejhfaeaebdiifbdbcgjdcgde
agggcgbceag                             
#geddjgjhechdfddjacbagdhcibiiiidgegbddfa
iigedifaige                             
#hgjfdjjgdgdhgjgjadijicgfghhgcjdehfiigih
fjfddhgdigd                             
#fdchgbjfbcheacgabcibgcfcjdiihejcecibbci
bhghgigijfc                             
#cfddjcbcedghdeceheaibggggghgbgjijfjhcjc
aiidcjaccbc                             
#afgabdgfaehgjigibdeedgiaaijibabfbjahedh
jcifcicdbdc                             
#diefecgeifgcididcedgcjjadcheahjgjejeabe
jgdhhfacbhe                             
#fdabicddieiejbbihgbbihcafbhidhafdfafbbe
gcihidgfbca                             
#haiedbjjffjgahhadbgeidigeeiiehihcfcgcjc
gfgecchecch                             

#acedejcabgfhagbdjaahadjgbaecdfhhcafhcef
hafhfcicbgb                             
#icfggehffgcibehciedjfahjccchghcjfhgjfdi
jadccdhfjag                             
#bihajfiibagiiefecjiehahgaegiaaeeciefjcf
fhijcfiijdi                             
#ciffgechdcggehajdicahiieecheeafbfbaifbe
ccafhjdjafb                             
#gcgebacjabjidjfjdechgiejgfjechbgabcaigd
fgijjhhbcfi                             
#aghfaeegcchiabddagbbchccichadhabjggeiid
bihdiagbdci                             
#gffhegefcbgahjfhhjhbgiihbigicghhdidcbgb
dicdfegfjgd                             
#ijedchgdhaiebfhacjgddcgjdfbidfcehibhajb
gfecagjejgf                             
#edgigddidhciidhieidggjibghheiaaiehdhgbd
abghffjhdjh                             
#heedeighbchabdibbfjejefajjidehddhffihie
hegfhgafaaa                             
#gaghfffbccfhdgdjiidicheedacbajicgeebhah
aeebehhaeia                             
#ejbbbhbdddedddagahfjiidfehhaeagffdifjgh
bbcfghigjea                             
#ibfigaacabegacgededcgjabdcbifgcjegbdfhd

ejaehaicacj                             
#afjhggffbaefejiegeedfgifffghffagjhbcdgf
hbeehacceah                             
#fhaicggdefejjadefebfagjdjifjgdfhhiecicj
bcdgiaihjfj                             
#jjhdbdeefbageajhdbcdgagcfchghgicfiegghg
hddjiccgibb                             
#aihidegecdjifafdgdbdeedfbbdghghhjjbibbi
adhjcfijgaa                             
#ebgjgieedeejedadghbfhhdihahebdfjjehaddf
feaghfhadbc                             
#aedaadhgheedjiiffeafgchecigbgfadfigjafi
jdciaeaadij                             
#jhicccjgiahdegdfegfbjiifggedhabcachbjae
dcfjajihheb                             
#gbafbiggjajhgfaegdghagcefdafcfihegjceec
djfgbfdicea                             
#ifcafieedgfgbaaejgggbcaeijjddhcchbfjeaf
aafchcdcjhf                             
#eacigigbbeafagdachechabicbdhjaaidfcfbea
hhjjhdjciha                             
#ideecijeeahbajdbdfjfbibahebagefjbdfedhf
jaibigfgbci                             
#gdjeaajciebfdefibdbighfdbabdbdjcgeahibf
ghgbifjbgag                             

#gccghgchcgiighcbffgfbjhfaaiebjfgiabaeab
fchbaaibebc                             
#bidjdjfibjeiidchfdbgdficbaecchheffgaagc
cjhbgadbdjh                             
#ccbfeibjcbfjgecafeibhjdbcajbcjiigcehdjj
cieidjcdbfc                             
#bbbgfhhjgjaahfhaifagfghjijefcaehaeighde
cgfcgeghcef                             
#cgceijbggafdhhcacajbhhgficgiejecejgcbjh
jahjgehajbb                             
#chbcjhjcagfbejdcbidbdgbbbfhciabbigabiee
ceaceeajdca                             
#dgfahhffeedgdcfgfibahdabjahhggafagiibbh
efbjefcebbj                             
#edgjcbjjfiahjhfhhjffabaagdbjdjfjiibdabc
dhgfjeaedjf                             
#hbjbbfbeeiejdhabdhehidaahgcehcdfhbbijdb
cjhcfaeeedg                             
#fidggggjhcaidjhgdhabdiafgdhibijjfeicfif
afdaffbcdcf                             
#hcgjgedcifchchideaddjaaieefdhgchacdbiba
bbihbjjacjb                             
#ajgiifaibbdbdibddicjeaiiibcehehaehjhbfj
cebdigiihej                             
#eiahffigbbjabcafdhigajdhcjgjagicebhaabb

bhiffceeabh                             
#cagdcgcebcicjaadjbgibchihjhgddefifdhcji
jacbjghggfe                             
#hghecfbcjacedbhjjjafbijeehggbiejjfgiehg
faghbdgcgdh                             
#gacghjfjjdbbjhddecdfjdejfdgjbddcbehjhea
fdgggagcijc                             
#aifjbhfeeechcbhbhadaeiaebdhgcebfccjgaej
eacadghhbhd                             
#diiibchffcbdcaihajfccfeaebdccaacbadfcij
jicccfbaeia                             
#fjgdddhbibcdegaggdedhagafiehjahafhhijja
gebhefdejdf                             
#dhiacijijdbjcefjbihcfjdefebdbebfegigbgj
achehfgegfe                             
#chiicjbiehddibehfibahighhfddchhhaicehaa
debfggjefca                             
#fjfjigcgfhdfedjhcfidbcjbjchafhbfcibdehe
giajgdjdhgj                             
#hgihgghebdbfadjhfhgaeaghegehjajjfjbcfha
gjcdiagiggc                             
#cfbbaiffbiecggahiachecaafgciifeefehheia
bafdfbhjaee                             
#hbfbihgcedacgfccfjgfjgcjbjjdeeiabjbidff
egcdgegaheb                             

#gdecaafcchdgggfdhfhchbdhejdaafcjafjebbg
jhfiddddbgj                             
#cfddbeabdbcfcigahadiiibcdifjhchieiafhbe
hjcfafdgdih                             
#hajdbfcidhchijjdbidjeejdcaejbaicabfhfdb
cjjeiiibibf                             
#cfaedjfdibfbffdahfehceidijdbdicgaebcfbb
jfhdjcidjcg                             
#jajhfhjgdjjihddjjjaeabdfecdcecejfgbcchd
edfidfcegfj                             
#baahfgfbaadcdadbdgfaaghcdajdcabgcjfaafg
hbbdhhccifi                             
#iaeecijiacajigcbcajffghaaagdheecgfgijbg
bcahdejgfed                             
#eggefiihhgdaaciefhfadffcaehacfaegcfgdjf
ahhaabeiadf                             
#bjcigdbcdjaabehhbacaefijgeagegedigifehb
bifhihdfbca                             
#cbibegbgcagecchcjbcaceccbcbhgjjefeiefhh
ecdagebjdeg                             
#cicbgehahceacihbcdgfghjjibfadbfebajgdga
jdefffhgfca                             
#cdcbeffgbgeibihiaiebedhhdijcghjijdfedfd
hhbfhegebjd                             
#dedfiejechacdihcjijbadfjdhigahcaichgghj

hfcdjijadaf                             
#hbhedhdebdbdhjicajdcdcbaggbdghhgghbdhgc
idhfjfahjjb                             
#ccdaebbeiadadfachghbgjjedeagjigadajdegf
effadibhiec                             
#ajhfgbdhfjhjdhdbjdcgffcgicfbcjbacfdehag
gihjbbgbjhh                             
#bgchjgagcfgaabiihbbbhijfiabegfcjbgggiaf
bfeajcegfbe                             
#ggjchajdbhhfhjcgfjedfcfcjcccgidjcfgcggf
ifgddgegaga                             
#iahgabhjciedcgjiffdachggbbffbfhhihcgcaa
aaafjbjjdea                             
#heicigcagdgfdfbaedbhchdfjafbaddebfeecgb
jbibeebjeae                             
#jdidahbhjjcjcbeejbfgegijchbebghfacifcce
gbgbjfejhij                             
#cfghbjagfbehegcfcffjbdeifddiadeccebhada
jcgbeafhchb                             
#bjijjigajaidcfhjbghccfgjhifccadfgbdigcb
iejciaibeib                             
#iheadiffdaddihebahgcchdcgjabdeagdfgdbah
gabacaajbje                             
#gbfagdcdjacdjefhgfeeeajdahghffeddeahijb
gfejjechbag                             

#hfjidabajecechiighdijeihbahjjbcjfghijhb
afjfadacfgj                             
#ahcicfhgjbhcjdeddcdfghjecfdiieffdiejeca
hjhcgihjbjg                             
#iejcfehcebbgfbbjjfejfghfaggagajiefgbjje
fbigjcjbcee                             
#ceiccgchfiggahcbhceccgeegbfbcdddaeffdia
fjjegfbffff                             
#jdaiijhebgfghigjcejdaicbieeedefhgdecchh
dbabacjdhgi                             
#hfbjjjibhgjfaghjjgfgheibgceagechfebcfgg
cdhecejecjf                             
#fabecifjbeecfbhjhjicejefbefbhbacdgdcbci
dciibbbddbg                             
#cigaaefbgeeihebddcciajdabfahjdccejfihga
cdajdbhjigh                             
#defhfieahiggffhhhbfejejcgefafgeiaecedgc
agcffhacefh                             
#ccecafhehhddjbffgbfbjcaiffjjgheaidijebf
ahaijafacee                             
#djhhcbajefddjedjijbgbafbjddihheajdagfab
ibghghhdbbg                             
#heahcebjjgjddcidfjifaegaffeedcddhcibhjb
eibejaaefcj                             
#fbbgfejgdjjejcabceadagfefebfddiidcjdbeg

jecaejbbcje                             
#ebbccegffcdbjhidcajbchehfgjfjfcjdcjbigg
aihecbbdbab                             
#jdcbihfaiebhfccafbaiidgihjcjbafeaagjjga
hecbafgfhji                             
#ejfaicdafgbhfbhdfgdgeccjgbaajjeeaahedbb
egehfbibibf                             
#hadjfchafjbdfacgfiebjdcbghfgghebceaidjh
igdhggiiehh                             
#bfbfeajahhegafgebfddhifdbeieafbdefjhafd
fhjehfffbfj                             
#ehddcbhbhaecgffibjiejagjjacafiajichfdci
adgjjcccdei                             
#jfgjahfhgiibdfdhcajiahaheehdhajdfcaaahi
edgfgbehcjc                             
#ffhibbfegaafiijefedcageiihhfhjbhfgcaade
hgcdegedigg                             
#gddjefbbfehdaccgibdgdbgahhgjjghbfheedfe
fghbbijccfd                             
#fbeaechihihdhacjcchbhdhhjaeghhjffdfihdd
hjbagaffidi                             
#chagaciihefaghggagcghgghgabgciifihcdbfg
diiigafcbbf                             
#hcbgbegigbegbaaedbbhbbdgaeheagedheajejg
gigjifaefjg                             

#ahcgjggachciihbfecafcfecgfgehabdgeahghc
ihcejdegehh                             
#caagijgeeedefhffagcieibhgbieehgaichfbai
bbcegchgfdc                             
#hfjgjagdjajeiceigddbghehadaabgaciefhjej
difabfiifjj                             
#dhdcdehchifdcffigjadggaadjcbiaghjfejhdi
behgcjjhbji                             
#ijfagiehgehjdjbiiffahbibjbahbjfifaeegde
fbhhajihjbb                             
#cegfaieafcifghfgfbffjigcfgiedibcdcghiae
dajjahegbhi                             
#hihafhcghehdbgcjcfabddhdefecbdbfcecdhca
dhibeejfdhj                             
#ghbaihbgjeahcffefbacgchjibcfieijggebaea
fihcdfebfaj                             
#cbcggchdaejhihcjfehacfdhgacjiahbcachccf
jjeeibbedjj                             
#agebfbghgdbabdehbfbebhbcgjhdaedijfhbcca
gfebefhfbdf                             
#bdefjiefdccbjabdigigjfhjdabdfebbjdagjii
fichaigbheb                             
#gcfabfciahbbiefdhbdcdfijdbhjcjcehbfhcbi
cegcibgicda                             
#ddifbeebhbgfdcfahdccedbfbjffdacdfhcaefc

gjbbbfcbhgf                             
#hcijhgijjeigaffccbehjffdccajiehhahabgec
bdhfbghibih                             
#jjidhfhaehijbdebgadfibdajhahcidcacjcdia
fhgccfigfeh                             
#aaeaidjbbggccihjjiedjaiiibebaeadcdifjgc
dadgdecijci                             
#bcjdaaiijhiifdjcijgfjcbiafeaaaibieeajbh
fffgdadhghh                             
#hadfcgjghjjeebabdheefcfjahaabgbjidbfacj
ihechjadedj                             
#faecaehedbbffdfbehdcdafadbeiejadhbcfhic
ffcffgceedb                             
#hcbgbjjfcgfjgdijahajhahbbadcdjhfaigciaa
edhcbheddch                             
#dcaehaaiagcdigbbhjagjdagbdeiegbigaffjhj
iebhebbbajd                             
#gaeabchehcjjefjafcgbafghacagihjjjiegbef
cefgbjeihei                             
#dacfjabeifdgggjdijjbhhagjfhjfcjfhiddcic
hjijdbjffbg                             
#daafdahbhdijhhaeacbceeefgfhfbcfcjaadiae
cfaehcecjai                             
#bieahffidjhbfffhhjfjgagddcbigbaihijhgja
iejjccdedbg                             

#acjicfcbadhjhigidgecaigfccejhghjfjjfgce
adehjahhdbb                             
#bdaadgehdcefhdeieiefdbjabcibgdfbhhibfdc
ahegjificag                             
#ejghadccaibgfhceacjiaaihgdidaibcgfegjee
cbafjagfbda                             
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caejefeeabg                             
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higabccjeai                             
#bbhiggagbfjbdhaigjecgaifhcbciahhjajbfgg
eehefehdebi                             
#hiajadjcaajeejfhcfehcegdhhcbcgdhadedcee
fgfffahjehc                             
#ccjfdihidbciciidchhcgiifebifcbedjajaigi
ifcdadfdbhg                             
#hbbhccggbefbiffafhdfiaagbjabahbhffadgdd
jdjgggceeia                             
#ihibdajgeabcabfjjbjdbdefajhieibbiicacih
bjfjagabaca                             
#abcdchjhifgcgegejadcaijacggchajehhadbjj
hjfhehcbfhc                             
#dhhfjhceiaigcieehfdhdecfcbjhedhbicebfjj
geadfjeecgc                             
#aedjefhjgiaeaajajdhfghedbjdejbdcaghdhab

aggfdjfiffd                             
#afeddffhfaibcibajcehbcjhiiddbgggefaagce
egbffbbdabe                             
#jggbdcfjggbfjaagdeegbhcdccjfjcahajehiij
djghjaiahgd                             
#dijjidibedjabgigdhbhfibdahhfahhheceggbc
fahigaiccgc                             
#hbeidgdijefigjgeecdebjaaecaedjbgcccaahb
hcggigfabdg                             
#agjaejegfbegjfdfeebifgiihigahbbcjhgchad
bbfgdceggdh                             
#ebjdidieaeedicjaaecabbcgcdigiigehbegfah
bjbhcagedde                             
#babadcaacfgijiadbgjhhhajhdcdbchifaehcfe
gfegcdhhdhh                             
#fiijahigbheeajeficdefcgeefaigacfbcefhje
ajdcccgehbd                             
#fjdghcgdjfgafbijejifiabbjecjidificdbbgc
fdehhhfafah                             
#jgdigcbbedjbijfebfeaegcgdbjhddafifejbhf
hbaibhifhgg                             
#hjahbjbbiafgjihageaibfcjbiaffjgeefjeddf
cajegfjgeji                             
#hbjghdabbjcgdfijiddjcdhabcbffiabebaegab
dccjhgjajce                             

#cghijcdijjjijffdcbibfdcgbeggigejbdahbdc
cjbhdcghjib                             
#ajhddhjeacaddifcgdejdejgbjhbaagbhhebcgi
icidijdhfbf                             
#achaiibebbjebcicjfgbgcficjeicebagfcgbaf
fahhijdaidb                             
#ghddadfijhiicgfdijahdhegacgidbidebcggjd
acjbejajdge                             
#fihhfhedjicjdccfecgjcjbicfgfiebjiefchbg
gaggjdecdga                             
#icbijfiifdgdihcgihbdabbgfigfjidccajgidj
hehdjcajedi                             
#abacacddfifbgegebccaccgcbaajhjcjiibdbeg
hhefehjdajd                             
#jjgihcaffaideefigjaeiiiagfbahhghffddgai
dcagedfiiaf                             
#digdaebjbeahjfddhhgghhgiheaffifffijejah
jfifaagghie                             
#caifgggfeeaaiieacfeghceebifbbhegefgijbe
gbachfjcced                             
#afjhhigihdfieeigbafjegacgagcdeighbibeae
jacaadfjjig                             
#ijbcgcjgbdabjcfcedccehhcddhghajcfcfejbe
baeccgeaedb                             
#fceffdaeejfheiafbjhcdhggjjjhabgjfdedfih

chdbcjfgdac                             
#ejfigcibchedhhdaacfdbgdccchdhiiahjchbjg
feabceeicge                             
#eijefcjabdagbaiagaihjhhidcbicbaacfejaff
hiecfgjcegi                             
#eghcgahgadabibgieahbcaifggejggjebgcjadc
eeffhijihgg                             
#bhaefbacichddbcbafbafbafaigadbicbfagjgf
jacgjgdffcf                             
#jjdcigffhdbcbajggbeddjcggibdjgdaiabaggj
aihidbhfaed                             
#eccjaehdbeacahceahbjhaacghaiachaedbcedb
aeeifjhchii                             
#fgjhefbieafhiebfbfgaiiciffdjegcdaicjiad
hijihahdida                             
#dhbahhdaahajaghadbcgdeegcejihhdhhhbehae
idbjdefcaih                             
#dbbcbadcfjcagdjicdjgacefagfghjijadcdgih
fiidaijegie                             
#hidceeeehcjgejgjjigbhhcghgibdijfeccddja
bdfhcjeejbj                             
#jhgjejhgcfdefecjieeegiagjgafbbiiidcgifb
cgggfcaijef                             
#fccgfhggbcgjjhhgdfgdeciegaajghafgbdgbah
hjjdcabiegh                             

#gebaddfggifddiifjeaahjjjcdbdfgbebdeheie
cibjcbjdfgj                             
#hhjhhighicagedaagidicbbdihacjhiiefdahch
biecghgaejc                             
#ggfadfcfbfbeheaabchjafjficehjgdabfigchh
ddjdjcjdjig                             
#cijhdehcacgiadbdfbijiaefdhdjieefheabfbf
icajcjeheaa                             
#bibjjehdfdjafeidgfjbhdgfeebdjhfajfjeaia
jbbfdehfcje                             
#djedjcbfcdgiajbhagdiabbggfaecffhjifbjah
hfbghccadba                             
#bjfddhjeeiadbhjhcbihgcdaahdadcfjggaffjf
jhgcajhaidi                             
#hdhdhddibhaeadgfhehifcciaedhjideacchjee
eiiigejacdf                             
#iafgfghhccfebcaihecjecehchfccccecbicebd
ibicjejcajj                             
#hcagbjeddccijejejcgfdacicjgidfaaidfgggj
adfdjgfbeij                             
#dibgghjhdfeacdciebcjieebdibaecgagdcebbi
bicidbihhed                             
#fijffhijhiaaaaehfijgafjedhfhchdgcbejjej
ddgajgffeai                             
#ehgedgahdehbcdghegcbdcdcfhedhhccghchgai

iefagjgfdhc                             
#febchaehaggehhbajffhgfiaaiigchdchhhgiid
fhdabghcaai                             
#gfadbcjahgcccdhadddaijjhfagjbjjehbfjdbj
gbeachfcgfe                             
#edfbdeifbiibgjbifjhghbibhdeaebbgecfjgfh
acifajciefg                             
#jhbhhadfhjccciibfchdihhcheaeagcfiijfagj
fjjfffcbjjf                             
#cjfiihbedghebcdbgeedjhafifibehjcjjihbic
abhhhafhdec                             
#gicbebbgejdfeighgfhhifccjiegccdajfjafdf
eiehecihdcf                             
#ibjgdbgeadifaajbgdcjgabfdfifgfaeahcficg
jhfehbjbfbg                             
#gjeabjiedbicjdciadcedgifeiahbfjgbhhigjj
aahgeagcdej                             
#bifjfejiaihejjacficcfbhdfhcidfabjcjbdge
jchigedcehi                             
#jhcjghfabiacdfjhhihcfdjfdhejddhegjaefei
fgggfbgidfd                             
#aedahjcebegcdjeacfcdheecbabiiejhddjibfh
ifabcajidjg                             
#edahbgddjbgceabficgjbahecbdgghdfedjccif
jjcjfbaiiih                             

#hghcahjcicaigjdbibabbdiejfgfadbbijjbbaj
gaebibceaji                             
#cdhejhfgegaijdcccijeaahidficbjcdhhaadie
bidjcjcjhia                             
#fjiadiebjihaccjbfgagegjbdaggfgdggjbjdhd
fefdgiggihd                             
#chghhicejhicghjcbjgjajddcgghefigddjbdag
eecgjdfegaj                             
#ihdfgeijiahcjbbaeigefhdhfbagbajachgjhhb
ebedcbhijfi                             
#headdfbibahaegbchjechfagibcfcefcfihcehg
jfddjhjifcb                             
#iaafehefceiefadceichfcdfbhbbchaifabbbja
cefedfiffgc                             
#ibhhbbjggahhhejhcjfaiejaiaefdebjafbihee
dbachfdchfd                             
#ccgfaiicadechcfeahfhiahbehagffdgjbgjfbc
gecfgchbcgj                             
#higjbjfgbajgijjfgicfjahefhbjdfbiaehggbb
bheiefjfjdj                             
#bbjiicgefefhgigddeebdjiajiabgcbhbbeieaj
heigbbddddg                             
#ghciibhaefeefbjhhecdhchbfiddbehadjjfiac
hjbdgcfadfb                             
#bjbjfjhiigcfihcgidddiebijgjaaiiaghhabai

cgjdgjhjijh                             
#igcdahigabdajcaecigjbcjfcccaaghcdiaaigc
djgfbfhdgci                             
#hiaajfabcibfaddjbjdecahigibghbjdhjgfahd
ihjigdbahef                             
#ajdaideigbbafdbbhahifeaigcadbfijbajecce
chejghbfibb                             
#hjcbfhaijehifddhfihibfaggecfehadgjhfehi
ejcahegjfgb                             
#ehdcibdiabffdfibaeefeieibadfccehdagabea
bgjgheefdji                             
#gccabggibccjhcddaahjjiagbaaicadhbfjccaf
dgbddjbhecj                             
#iecaaccdafhjacfbgfhcdjchjfefhdciefbbdij
ajfdcgjdidi                             
#jhfgaafbcgjajacbcicfcccedjccacjcdhbidff
abddbbiahcb                             
#bejghdijdaacdeibijicfjcgjgaciadjcgjafdj
egcdeadceih                             
#ihdcciabejchhgaafihahffehfbcbedjgaghfhc
bdbahdjdfch                             
#ejebdibbdjbbidbaebebabbbaigdffbghijhibb
dbfcjcebdfg                             
#afecgfghhcccfafafadhigjcgbddchaadhihdha
aegjdfjfgdc                             

#hehhegjiefebbgiaghcbbfegifhhhbffaccbcbf
gbijjaejece                             
#jchbajgafbjedjhfjhgeeddbidgaejdabaicccj
ghcfddjecge                             
#hcjdicbhgjeadcejaeacjdhbdeaidedigehejcd
icebjebeieg                             
#haagbcdghjbidjjahdfbdgfibjideihfidhdfif
jhhjibfaghg                             
#aagceafcbjfhahjbhhaihifbfagdfcbbafjbiae
gfccfhficdb                             
#efggaediegiidfcjgdfajdjediicjdfcbfiacag
ddihabbeejc                             
#fhbcbfggajaiabcdiifajggibdhcfdhfgddeieb
ffgebcjehac                             
#gjbejdhjcfbdfeehbbdffbhiajhihjcaajjceeg
hehfddhjhhj                             
#igdjfehdifjbiihadefhbaehdafgabbfbagdidh
hccdcgbdjig                             
#fjdgiajgbdcfchchgjbfadgeigjidigghbaedgb
aiibifaeddc                             
#iicceebihbhaebcjgeihjijhhhjdcjcfjjchgie
ffcbabijjcg                             
#djbjhjdhfiaajhecghiabhheedghcacbfcgegii
agcdgchbebg                             
#ihfgeahgighbebidejfgbfegjeecbdhfbhcbfbf

efgaieigfcc                             
#bbbbcgbgajajgcdcdgibfgbifdaefedfbdggced
hbiedghjidi                             
#fcfhjdebidihcgdbefdiebiegeaahjjdhfdcaij
icihebfcbgc                             
#bcaijeadgiiabeadfcadgddhhggcjjbbjdihiff
gjieejadgbj                             
#hciidgaiifjfcffcehgecefgcghfbiifefhfiej
bffgjiedfif                             
#afhdcahdbbdgbhdedeibjbddfjbcibhifajgjbb
hbgajcjceag                             
#bgchgbgfhhbcdhdbffebbebjffddjhhggabibhd
afaehjdhhfg                             
#ddacccdcbeaaedjcgaiihhedhcfcigdiaddjhfg
ifaadajdgeb                             
#fedhajeibbciadjbhagaigbibihjcjehhgagicd
ecbebbjjcbb                             
#ejagdbbabcdbddiifbgbeighiiidjgiecechibj
fejdaiadaac                             
#eigbjgbahjcjgahajhjdhbhfehdhabddjigjcef
cgfaagfbihg                             
#cfhhbagjheafhjiceahdibjgfdaficaefacachd
jeaaecbdibh                             
#fedjidbfbcjadedafjacccdffbbdfdgiebagbgh
jfebagejfce                             

#abbgjchbihiebagidecdfebiiddecbjiecbfdca
ajhgcaijehd                             
#fgcehghccefbdbhdegijbhahijecbedbcfgjihj
figgcefagij                             
#djbagbbccbbfebiffjheccdefgjhgcgdhidedgf
dgcbeibbhfc                             
#ghjdjdfdedchabccdaajhccjddaabcachhffjej
igdegehfdea                             
#cijjchfcbfbehecfgefcafggcjcafegbfjffafd
igadgajdejf                             
#gjhfcahajbhgejcebjhcgcbdabdidbjibeadehf
caihgfjbiif                             
#dehdeciabcaefcfahejaagjjgdhddhgdhdbhbih
bebbbfbdcig                             
#ahbaeaagcbaecdbhjdajgahdahgigdicgdiiihe
chiadfcabib                             
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gbhaciideib                             
#ghgeiggcbibchajfiajedbfbjbjibjijjagfbcg
dcjhbbdjfbi                             
#efjiaibecagafdebbfbchjdfadbhccaihicjjcj
cghhfdbafid                             
#fdbbhgfciajbcdehdidaiehejciijacfhgaebhb
eahdjbbgjab                             
#ggiejdijjggjcbjfgigbbdeegfgbfdidgahicbi

ebjgeagfdjd                             
#bjfdbecbggghfeiechgfbagaeadfbaifjbadedf
jhffbfhajaf                             
#cgcbehddaiebiggfjedbgafddiafibgcbbhjdga
eaibhbbadia                             
#eiefcgejbaidifgbehiabdefcjbeaffbacggbij
jhhcajfchhi                             
#gaieggcahdgabgffjifgjegfhcaeafidjhdhgjg
ebgffcjbfdb                             
#gacffhgifdgjdfcefgfacdeihdgiagfedbebbeg
hbbaajjibcj                             
#iieddgbbegeaibcjgjfhfcabdagejfgadhahiei
dgchicdejgi                             
#jcigfcdchejjdiaihfbadhhaceffigdffagbdcg
bfbajcgjhfa                             
#jidiffjaecibjegefbghhcjhhdcjhabfcafeghg
ccbdbgiggij                             
#baiedibaeadjbcffhhajfdahebjaigcgjjbbfac
dddheceddcc                             
#gfgjaihhaaiheicgbcgdbhbebeggagfabjhhhfb
ijeigbiiega                             
#debhbeigiddddcdebghfcaiahfiaheeiibcdgac
icdegibebeh                             
#adghbaaaeeajjhjgjaadebjbhheiaifbgghffhb
hfgiibecfgb                             

#cbcjaghegghjegjjeabgfaeciffhghfaddhaihh
iifdcfdgefh                             
#ccggeibbhiihfjdjdieggibjfcdbbjbdhdhcabf
abhegacdhch                             
#gfafddijabbagfaiehdaheffefeaaaeebcjgjif
bagbjigifgj                             
#gbjdfaegjgddaafhiafhcihidgdbdhjeaccfbfa
jbbciddabjc                             
#ggdjdeiedfhaiicdiaieddidggaaihbagchebij
dghhegbfbhg                             
#cbhdjabgedjheibbjfjaebjdbgecdhjhheffijf
dechaccfjfg                             
#dbbdhgibebfchefcgefjaaedbaebacdjhgbahcc
dhagfagaehg                             
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fhhghffecif                             
#jbahgigebeggjddibfghfjaaagghihgfgbjbjjj
igghgchcfii                             
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ggdcdihfeif                             
#jhehibechdhifdiiieifjgjbijjijeedbhdgada
jehieeadfga                             
#edggibgccaebabdhefdhiaaegaghgbbgcgdeijd
adgabaiecae                             
#bidddffcbbcbjcefjeeehgcggeffbebachdfgfe

ccjejahjgaa                             
#gffiibijecacggdbjgjiebchbfagccdahahfbgc
aeffjdacbbb                             
#ehjcbcighbgighcbifajdhbebiafifaaeggdcbj
cgegiafcgif                             
#djdfcegjigegiagaiffdjjfjidfijiadjjggejj
hdifjghbiah                             
#gecbibajhijjgeddbbigiijhhbbcgiibdbahifb
eijfdcacife                             
#cdjjbhhffhgbhdhhhcheeafieigjfdedefbhgga
bfigbfgecgc                             
#hifaehhfcbcjdjgedcccidafcachdeddjjffhde
dafeahjhbah                             
#hhccffhigjccfadbbijigdbhbdceefjeeeghfgg
cbgdhaecgdf                             
#egagdjghafdjciafcahdcidgfhdieiaigjbddbj
idiifbgefdf                             
#jfcbagjbiedegjfacbijjhebhfjacbdihcfiieb
bibcfahcaad                             
#efdcdfidjigjhdeajhifbcedghjegdhciiibihe
hfghbbedjjf                             
#bhhgffcbhhjdhfgedbbcedahddcijdjecjciiej
gfdcdibhdig                             
#abccdhcgfceiggibgiehhbhgbgdgjbchjebgfbc
fahcejfijih                             

#ahbcaadcjeaibbghfciejajgejhicfagacbddef
jjjifhhcgie                             
#hhficdfdbaefhbdijfdhddiabhicbajfiaeciba
faahhjeijaf                             
#hjbabdjeehcghjgcfahibgibhdhejijijjhgaci
iiedibdbbcc                             
#jjhfggigigjbieegedhbedehbfejgdhgjdechhj
gfhagjjjbgb                             
#cehaebbggggahggddfajjghjjhjjefhaibbedie
cafcbbcbcfd                             
#iegfefhbihhdgbciebaebbhbchcaifidjgdiadd
ibcdihcfhhf                             
#cfcchiigjjacehehhdhhijcbagibgibhdceajaa
ibddihafjfc                             
#iffbbheidbhjicgafbafagicigjgceiffbiiijj
dbfgeffihbi                             
#ejjidgfcebedahediaafhgfaiahgddijgeebfjb
acjjgjjifaa                             
#efbigjcbajcjfhgiiighajbidcfefgiajacicbf
fcfccedcaab                             
#hbhdjeebhajjgiehfehdaaigfaijdbcihhcgcfg
bjgbfagedhd                             
#jhbhfjjigbjgegcfjfjabidhhddhjcfadjbhfee
gggebbiehib                             
#ehjedahggbgicjjgjdcabdfabiefffcfiiebaah

ffceiehidij                             
#gebjhbggfdibiiiciicdfjheaggcideihjcjajd
didcfadfbja                             
#gdcgfadecjhcjgfdhhjagbhffejdbddfbhehcfa
ididbbbagdi                             
#fcfgccgfbbbdbbdcjciggghhfjceafibhjeggfb
hjgjdjjgjbh                             
#fhhdacdijcajjgadcbebabjabfdiijijhaaebgc
igjdheebefd                             
#jifheigjdbchhbicagdhiebbibaiehhjffcfhfd
ahcabahhejf                             
#edjighihihifjgacicbjjbadjifhaaghddcdcfe
jfhbeebcede                             
#dejfeehbcaajefjfgeffihcfgebghibeafacfbb
jfdhbacaecd                             
#eeegcifgjffhhjbbbdiehbadbhcehejiajegaad
hdjjdejhiba                             
#dddjcbffdgdchdibfacgifadhfadecfgehafgid
eificfbgcgf                             
#caaiiefecbfceagjdeajjdccibbbhdidhbfgcdi
aagibjhbacg                             
#gdedijeedabdecjagfhhdgihejafhigghiggbia
gajffjghgji                             
#hcbhadfidghcabeehbiifjaeciefdbcjjghgigh
hahdhidghdj                             

#bdafbhjhdfhghhjffhhdgccjdhbaddhfebdhggg
cbcadgefiif                             
#hjcaecbcfbbiajfebfjigjejhjecbfihgfechbj
ddecgcfajje                             
#cbfabcehajfffcfedggaedhigfdgbgcjeaffhai
hacachdcffi                             
#ddbdehdhajciddffbabigfbgfighffjaheebihd
jafeiahiigj                             
#jgijcbaccbjejicejichgiiafifadjgfbjhcade
haifchffibb                             
#bjjeghdchhificggadhajjejdhhghjgffiiddij
hidifdibjjg                             
#cafaghbjfddibjbhichfhjbgggdagecfeidieii
fhihegefgbb                             
#fiecifiehddacbcgfjjiejdfhgaiiccdigahjed
hdgefidjcfh                             
#bihbdiefhjceeggfigdeebeajhafhgabjgbiicd
idggijehdje                             
#ehfajjidchddbfjeedideifddcgegbjcicgigja
ffdfijhhhif                             
#fgbagajhbdbhjedbjdefafiegibffjbfcjdfifa
ccdbdaaieji                             
#iaehgdacdbafgfdaiafhbcfahaeagbahjjiafch
ffdghacajai                             
#cccijcbebffghhdfgbdgcdcdjijgfjejdbiccbc

bffghcihbej                             
#jjibjbjcjgbcgihjgefcbfhijfhibhiiegffbeh
abjjhcbfgda                             
#fciadhhcdiidbffcjbghhjfbihbjjgdijbdiagd
cedaajhhdja                             
#eiecgiceeggdebhjhcbbghbdcacbjahaefifhbf
fhcbfjdebbi                             
#icbedjgihbgdjhbbhjbcfajfdicddhbebfjafgb
aadbaebjgda                             
#baihdffhideejdaaehjehajiecheibgcbhhdgga
fdedijaihid                             
#jecgcggggcbbadgdbdeiiehigfdeahfgehcghfi
biaajgbjedd                             
#cgfifhddjjdgbgdeaifjihcjcbjcaidedaeedah
hbbbfefegca                             
#aiieaijahaifcbjjfedacajifbaieibefbdgdja
bbjhgjfcbeb                             
#chbhcgdhhadefjjibcdjadfbijihjbcajjjaecf
dabgjaicbej                             
#gdfibgjjfiecfdbbjahhcjgbghbibejifaihadg
bhedfiejjbi                             
#aaahbcbgjdafidgbhebcjecihebbhgihihcehfa
aijcfhjhecb                             
#idajhgbfeeejiejjjjihchbedeafjjbhgihfgie
aghhbbddbac                             

#jcggfcdcfbdhefdijcjdeeddfjfdhbbagiijddc
cihjbaifgji                             
#jjejececcafhehjghajdeeifdahjdjcfcafdfga
ffhbeccehgg                             
#cheadjiiajjgiihdadfcfcffbchcbfbcgihdchi
hafcdecdgba                             
#hedbicgibdcfiachhfgagfbcaegecdbbabfhfhf
ehidajffaci                             
#jfjecaibhafbicecdchhbjffgghhhdjifhfddjd
daaedfecgee                             
#dhiafbjigecihdfchcbadbaccbdjjebiaabecbc
idfddfjdgbb                             
#ibhcacaefhaeagdbjbcfihggdihjiebdjhjejae
daajjhcfjii                             
#diicabcecggiiidcbcfhfbfaaadbjgjgehbbfcb
jhebaaebaeb                             
#hcjhfieiciaieibhfiegebjjdhfefddghiiijaj
dafggaiedjc                             
#bhhdbgfjfgdgcjcffbbibgggaebgbdgciejjbhc
hjbfehcghhf                             
#hihijhhgffiahebiagdiihbgegebichcbbgideg
hhgfbjabaja                             
#cbgefcicfchajcgbfihagicfigbcacicgjgebfc
abcdefdhace                             
#bhcehjfdcgfhjfaajjigihgeajbabhdgejbaidi

jibgcgdebdf                             
#jdaeehjihfigafejiihiahbhdajeeddfgdicici
edifafjjafh                             
#cjfiefiebgdghdijahijgcecgjcehfbbibcjfcc
edhgahdaihh                             
#ajjgbdhafcicbfjcddaadaafhfhfidbegegfjee
gedecjhhihd                             
#cbbiaifcjdhbicabciijbfadaahifihdcibidai
aiggicjdjid                             
#afedaihddhahagjehiechahebbheaicffgcjacj
hfhbidjehhi                             
#ahgihbafhbgcidhbciajfhcbicajijfibjbcdae
chifbbciigh                             
#jgahjffbcehbaghhggjccefggaiddecjdjdihbd
ifgfbachigc                             
#gdeafcgihgeidefcabegaebcadhfdggfhffdgfi
eccaifebeca                             
#bffdidiajiidiaaefcidehbedfhejbcjjeifhjc
idbcjaafddh                             
#cbidcgdbibbadghjhhagiffjdedihdbhagihcih
bafjjgdjeih                             
#ijjchhdfffahdgighchbfabaihgijdfeebcehgb
hfbgafgdahi                             
#aiagacgijhgfaecjceeegjijfbdfcfcieajeigi
gbhiiaiibfh                             

#afbgfdhhbgbdibaehdfcbgcbcjaedhdcbbgiihd
beibfafheec                             
#fdcjhifjbebedgjbigcjcajhdjdeejafcjdgaab
hfbggeiidif                             
#agfefiedgadicfhjfdiidggdhhcjjgcgbhecfei
ihcfcddabef                             
#egjhhhbhfbegfahahbbbaibiiejgdhaghchfica
jhfbbbhgdhf                             
#ggeehiafcgfdghcbcjhhagbjbggbdjgcafhfaad
eebceaegfeh                             
#eadgbijjabihgcadbiejbbgfbaiigaahhfahfch
jhciececaca                             
#bgddbhijbdfaahdgdiehgecbjeddchjgaheciac
geagahhefie                             
#iiahhaejicfceaibbcahdjhdjddbbdgbccaibja
fihhjieifjh                             
#dahbiaijgihfhhabihahafjbbhididifbcdabcb
cagcjfihehb                             
#bhffchjbgjbjfagcafjeeajdjeieafgfheiehfd
cfjjjcbbhah                             
#addhbjehgaggjhfehbjggcbbdiedddcbdbbchhh
chfbbjgabhg                             
#idfeacjbfhgcccdfichbbhhfdgbhjgdhefadgdg
gjjhhbfcbic                             
#jgaiccjegbbdgjagciiiididheebibfgadjfcfc

jgejedjeiah                             
#fcagdahfjcijcbgbhiijieghbhidhbhbdbcaaai
efjffieeegc                             
#gifbcciejabddebbgaibhdceafebfiigafjgdgf
iegjibaabji                             
#cfaciaghchjaeaebihjcebbdaedjddajdjfdhij
iahhagihffe                             
#ajefhfiafcifdiaciffggbgbfffcdadaffgfdgi
abjgfgdieca                             
#bhfbeghiiaciaihfhfdjhfaeefdcfiicgehjdhh
aadbjbbgbgc                             
#dcfcgcbdefiefjebhjfgbiddjagjiaihabjbacb
bdajdcjcefc                             
#cfabhjggefhjbcabfedfeafhfhdfeaibbdbhhgg
ihfhgbbjecg                             
#deeejchjjjbgdbgiiacjedabdcagbibbedchhge
hcihehebfdf                             
#gigfcbiggejfchheegbhgjddedieehddgagfieb
baeijedbbfe                             
#jcjiddcgaiheijgacdehijjjhabjejhfdfdgfja
jggheechbeg                             
#jhcfgdahgecjagfbdibfdabgdcehhghiicfbffc
ficadhjhbej                             
#fcfgdahjacfidgfejheadbbffecgfdiecebbcah
bjjigafgcec                             

#cabejgfhghbgiabbjcfceghgaccjjijjiibciab
ccebidhdbjh                             
#eaidjjghhdcdagheifiijejfihdhbgcaeibbjba
jeeebihfaif                             
#ejbdahidhbcbcfcgbigbgjcgcccegbcidbfajdd
fbgbcdggchg                             
#ehhdhgdfiecghbjiigiifcjghbebdbhhgccbgcg
fjcgafbfcjf                             
#jdbidbedejjddfjcbgffghgbdgdgfjgeihdcdbg
dbdbabffjjh                             
#bfhbgdcgbdfjgcgdjdiibegecgdjbdhfceajcae
ceihcjjjjaj                             
#ehhaihjgghdeafjgcfaigbaeacebedgjifjgebj
dafdjgebajg                             
#efgihecjhdcdbbagjdjdaehheficiheaejdfeei
eiadiegifbe                             
#bidhecccaabbgfajdhiaeejfejdfegeehfdbegh
gghhifcgcad                             
#iieeigbffdgifghbffffhigjajihccjchbgjbca
dhffcahbhab                             
#eigjfeaecbiigjjijciecfehaiaigchdeidcaed
geeaafihdai                             
#eaefhddbihjgbbfacdgafdbedihhdgcadgfacfi
ichigecajaf                             
#ejdiadgbajihgcaefhhajcajfjaghfcbjdeeiib

affbhjjejha                             
#icbcibhgahabgjefjhfcgdjafggejbfajgjbbic
gbhebgbegca                             
#fibdaeadfcaccigbgaihhbbcciidbeeggfhjhga
deeaeaghhbd                             
#fjdgdhcbcffgbfihfeiajafciaigdieadeehjjb
hgiihfcaehd                             
#dgaiajcdehbcgcbbhghiigabfedfiahgggafiab
hhiijedjffe                             
#eiiecgihheaabfebheifdeaggdiifejabdbhaeb
cdhaghejaah                             
#hgbbafhgfjhebiiiifhhgchifdgcjhajgijacfh
dhgedcahbag                             
#ibbihehgehjgigidjiaafhedhhjbcjgecdacjaj
hefbgghgdfa                             
#gjfejeggjebfhdfgdhfhibdgcjcfdebbaeeiejg
ghjdhaajigf                             
#fchfbjejgacjajbjjddhhbhfbgfadiidjgeihgi
jidigcjicef                             
#cjcgffijhechgdacbidhajigihigjfibjcaefjc
hhfieecdigf                             
#addeabaciigibcfabbbiidiehgdcbecahieibeb
fheibahghgh                             
#afjcbbgbhejcigfbhacfcchgjdcahajibchghch
chhebhggbfb                             

#cbdbjbccagigjgchhhdiegiciegeaiefibggagi
hieeceheche                             
#cfgbbhafddhghicggjgggajjfjjihdcbiheehcf
cejfghaajgf                             
#aedjbjcgfahechaadbjcefhhcdhhegfdcdfabic
jdefgbaadbb                             
#fjfjeecdjbbahbcjadjgdhgfgjajehgcijffdid
feecdgbfdgg                             
#ijiafdgfbgifigdgccfgcbfaaaacibajhicecff
afhfgdhbbaf                             
#ecgbjjbebihhfejhfhifeffbiaeceiifafgcigd
efheejabcji                             
#jicbebjfhdfcdcgdgeafjiejfbbbbgbiehdfbif
bedhhcegcbc                             
#jihagdggaiejhagbfaeeffbfaacbfihbejeedea
afgddbdbdfd                             
#hhhgcbhfgcjdfgaigijbffhfhgechcjhfaceccf
gjdabfbgidi                             
#abjgfhhiagbfgjbjihcgdbgjjihheaicgjfhghf
bfjifedaeei                             
#igibdifajdajchehbiiegidjcfjacfahcjicgii
hjhfjajcfhc                             
#gceijfejagbiabccbdehhbadijjaidfcihaeebi
eaeiiedjdad                             
#fejcccdiicejeecahiaefgachhijdghhhggjfjf

bhigfjggafh                             
#eaggcaacedfdfefcbhccdagiceechcdacddajjh
bfabghfjefa                             
#ffhhfceffdiafhcggcbcfajgfjefibgfecaggfa
hecjbhhifcc                             
#aiageiediccfjedcfcecfihiciheabibdcgggbf
eadifgjghha                             
#bggebibbgiicacjijhdgjjdjdbahgjjhiadgage
dccijjajfee                             
#afjgibiiicbafgdfdjjidhhbjdbhhfgbfbbeefd
cjgbgbadbdf                             
#didhajggfjdghbiejfhgdgehdicbghicfcgdbbj
bfefjebfdga                             
#fhigfdghdcgjcdcjhjeaibfhccdgfahjeffcejf
edfhgbbbaei                             
#ejigiabjiecdgeiafgccijceadbffiaebhjgdda
gjbchihejdb                             
#ffbhddahhaaedbgdicdgfddfbjjdeciffacdfah
aacaebchdei                             
#iffaichbghgdefjbaihhachbeadabcdjjchfaga
ejjjdegdgdc                             
#giiaaiccfdfjehcdbaigbbggghfjecbchbhfaej
afieabjjddg                             
#gdbdihgcacggehdgejcahcaiaihjjbiicdegdig
edaiaeghdje                             

#fdggbdhbghidgggabaajbjgebdcibjfdaiiighc
afagccccgcb                             
#efbjjhhgaeiibhbcbjhgggabgccaeehebjdiibf
cgfjijgbdgc                             
#gdeagcjjffdciheifhjfjbacehdcehbhcjgahie
cjgjhjhbgfc                             
#hfbeedjjddbbjdgijigcjhifgijifghcecfdech
ggghdfgbhih                             
#chddacggjbjfhiehhihbihafgdafcdbifecichh
fhgfjeefjaa                             
#icicfdbbbgaijjegcfhggicajfedjjaaiebgdea
gdbjhhagfaa                             
#faebiieadabegbfcjdhagbhbiffchefheffcefc
fgcigicefab                             
#gdgdbijeijccbdeejcchdcbfjhffhaaaibdbhcb
icaahhhfhec                             
#ebfehfeecjegcafbaajebcbbfhieihchcgcacfi
bbiadcgdjjh                             
#giedfjefijiejehehahahjjbehcgcbchfejbaab
bbgdhejebaj                             
#igfeeiicacfeibiijejhajfajcfjedcdfcggehf
geiifjejide                             
#degdbhhjfjeeahabfhfcffjfjcichechigbbcjb
efbhdeafjbi                             
#eaacgiebjghbhcbeaigiaibfhdhbadiedgijeeh

bfhbiecjbhi                             
#baihgahggbcdgdhiaabjeahagefffjjjdjbahdj
acgffejfabi                             
#bfecidaihcbjgbddfibefjgebbffhihgdifgjcd
fcbeccejjfe                             
#hdacgbbheeibcbiehbjhajhfbcbhdffgiidjgic
cfjiieeefhi                             
#bfhjeiccgahfagcfdjfgfbaghcabgjeabidjjfd
ecffigjfbdg                             
#aggeddajddehdddebefbfcejiagdidfccgibcda
bjhhiigjedb                             
#dfageddcjgiafbiceggcedjghcfdjdfaedhbddd
iijfcaficig                             
#efbgfbjbijdbabgdihbjfecdacgefgdhfdgefca
fdieaccfbbi                             
#egciegejaaejifjiefiagacdefifichdebdhfed
dhdbadiahbj                             
#cifcicccgfhhcbghhffjacabgcccbbddcidbggg
cejdcdccbcd                             
#jjbhafdihbcgjggbdbgadjiaeechegegcabgfcf
higegfibfda                             
#geadehaifciideefecheaefdeageadadjcgfbac
bgdaijahjge                             
#eajdcfgibigfibjcihfjbceaiieicfeibhgddjd
baajdgcjbag                             

#hjhdfjhcjcabihdhdhecagebaahcfjhhbgdcdbb
agheiifgdbi                             
#jiegicibfjajbgcgieiidaecaicaeiiagfabchf
edhbifdfcaa                             
#fbghfebbgdhbbdidiiggfhbdeegagieeedbefae
jibhidfegfe                             
#igfifeeagegbcfgcgdbadegadjfbjjghhijedad
jhghdeddbfj                             
#jhbhiieachdiehhaddicbjaadgigjchgegdcchi
hdfjhjjgihi                             
#djhchecfieeaefcafbbgjffhbechfeejjddbeba
bcficchhcig                             
#eidhcbaggcggfhfjjhfgjbgjhcbdhbefgfaaica
ajjdchheaih                             
#cgihcedaihbjfeigfafcigadgibjjbebbafdedg
ehdjcbfefej                             
#hijfhcciegffjfddihabfagdgddibcedfbhjgci
gfbigdhbidc                             
#hiibafjhbchfhggghgchebfhfbhhagefdicfagj
fgjahdjdbge                             
#ahfiafbhciefhdgibbccaaeieiefieaiddfedhg
dciebibdddd                             
#ghjddjehfehcjghhabejiefajaajbbfcjicgcjj
fibdaifjidd                             
#hcafgeehbbjcicegdicegdcgfgcgfdihfhefebe

afehaibdbfd                             
#jefeiijfhaffjcbajegggehehefceacejgiiefi
cahgejbbghc                             
#ebfifejieahegegbcbbhcibbejbacbjgfjjcaje
dceifgfchej                             
#acbgdjjifchghebhibfahibiijfabcfdfjhchaf
biieadhdeaf                             
#edeiaaiheehbdabhhciihgfiaefeijhacffdjjh
hadjbccdcaf                             
#bajacagdihgcbcihjfahgbddbjbbhajgeccbgha
hcghfbjiefb                             
#ceihfjiieachibbeibjcbgehgfgejhbgbegaeij
cgbfbcedfha                             
#dciicifhiafcacbdceebdjciahhhgifdheicjgf
jgehdhhhefa                             
#dgjaifgafichaedjadieaafigcgbgebeagdddea
fefbfhijcca                             
#hbjfcgfjbaaidbihdjcccbbhebebfagdebgbdjf
hjhihgeajfd                             
#ifcacbjgecaecgjdhcbdjgjhcefbghaffgiijcc
bgdfhfgifdi                             
#bbhgefffidieehdbacedgcjfcffeaceaigheded
ejjdjhbdjgc                             
#diabdgfajhheieahjiafdcbdfeieihhggeibfcc
cegdciggiac                             

#aaeffcadhdjjedgahgjiicfcfdfbfbajfhbadcj
jfdibhicgcb                             
#dbjaahabhfajjjjbajecgdiabfafaeejcaifada
bcdejfcfdei                             
#dahjbeieibfgdbdicddbibjijibbbgcfefeeefb
cjbchbcdgjf                             
#bcjeaaajdgifgcdahigfaiadcbfiadhhggjgich
gccbjcdecbb                             
#iebfeaihbgdagibedffjhfeidffgjcjadjgfcfe
idedjdfeige                             
#hdiffbbciicaacciigahbfbfbejbeidfjiajjgb
cbabchjfhah                             
#acdgefgbcjjjcdhjdfggfcgdgbfieddachjibif
heidfhdfefc                             
#bdiajcihaidiabagidjhgahibiidfeciiaeciib
facdbficbgc                             
#eifggfcbjbjabcdajaajcddghjejgfgigigbdgf
fgdaaiahaha                             
#haedigcaifcjjchgfhbicfdihiceffagijgabde
ebjajcbhddf                             
#gchdihaiiiiigaafiidfejcdjabffjgcgceaefd
bjhificfdad                             
#dhegdiceecifgbdbhhdiagbdhciifiehgdcbbjf
gigdiffjgbg                             
#jjbfahiddadbfdgffjahhdaiijaajjdcdfdcgdj

ejfgfgfijbd                             
#cgafiebbfiehhadaddcehacabjfdejifbbecdef
eiagiiaefif                             
#cfdighaadfcjcbcgdcbbgiebaaedghgaehgidab
eeeaheejehg                             
#cbcciccadiahidiaihcffgbihabjjbbigaeihee
jdjgfdeagij                             
#adfgibfbgbaddegdehgbieciadcfffcfafiajif
ifgjcbhiefa                             
#fgfagddhbbjfgbgdhgchfgbdbgbgieagghabccc
gjeafjhihhe                             
#ibifidibibddddigdbbhcjciibgfdgfhbhgeigc
egehajefbhj                             
#dbhiccaccbejebdgjiccbeihbgiabbhjbdhfbig
jgjfbidchaf                             
#diiehgiiibcgcehdbaggdbbbcichjhbbgaiabgi
ehjabjfhbej                             
#fihjgbhjdebfbbfibffeeaciaajecicdjjijibi
ijeegijgjef                             
#jdcbaebibahdbejaeabicahgdabcfefiifcgejj
fddccbdggbd                             
#bhhdabdcjdihiceajcbdgjhghicbgbhcfehabfe
jdeffbehfie                             
#jfjcjdbffeiibejgedbbcjbdhabddbafifejide
gabihghhabg                             

#bieagfcbgibdjedjbhhifceiahbhfdjjbbggfeg
ajcjaaejdbg                             
#hacefbicbdjjaegjjbjgechihaciecbedddeaeh
ffagibjdfab                             
#jacecjidfcicibhbgjeheigibjeifccjeagbhjc
fedibdjidfg                             
#gdbggeehiihfdgahjgahhabicccgbfieecifjij
ajejaaedhgj                             
#hihaggfehbjeabjbbhbadaaefcffgdhgacfjabc
hbfjhehgfde                             
#cjajfcefdgjicadhjdfhhehaeigfhigaigjbbbc
jdgbeefgbjj                             
#cdfafgciiiejeechdgaggdcaghfgdafeiifdhfh
ghfddjjedhe                             
#ebgiaddegdaeedjihegfddjdjgjgeeajhbhhaeb
cibbaefhegc                             
#figgaedijjhfcegjhjfcabbejejiaiffbhihfje
dgdficicahd                             
#hbdhaaigdihgefddajefeajfcfcfiihgbfdgahe
ebddjieecif                             
#bcgbhaafdaaieghjbddcbfijdffhghdfiddefec
fgdfgehbggi                             
#cibdfgcehihgihebhcjgdgaiiefdidedebfidda
ecjhjcehefd                             
#ijcejjbaebigidbebfdcieieifcihiaidibafje

iigcebehbac                             
#bfcjhjijigacccgihhhbaceahbbefjbjijfbfdd
decjcjcjhad                             
#hifhichgbdgcjacccideijgbjghjifggbdgidaa
fdijciijbha                             
#icaaeaefbhdigjbiajjidacfdaahiabhebiacjf
gefiajigade                             
#fiaebajgghijibdadccaggadaebidjfdedcbicf
dcaeidhjbii                             
#cacdfgbijahdcjacijcdddedgjigdaceebaaaid
biihghebgaf                             
#cjadecdbbchebacfhahiffddedacdcecdfbgddd
jgibjhdeagh                             
#jfjjihidjjdhefhciffehebhbbggchcbbdaadaa
ehebcdedfcb                             
#dbeeiageefigfeebdafegfdhcidjfgjhchfaggi
djiggajccff                             
#ajagfbjiegbddheacfffigdcfaihagdibicjdji
eabdeifcccf                             
#ceaafghgehhfdhdddiiehegfijcagdfjgcccabc
hjdjejbfbaf                             
#daebbjgbjbjiahegfbchfacjbdhbjjfcdfieheh
hbifibdaeah                             
#bbegbaffdefjdgieggcgbgaigjcdhcfbccegjhc
bijhdebacgc                             

#jdcjjcchideifhggcfhcchfghjigfifcgjfahed
ehfbibfcdgb                             
#diagjejehdafgabbhchgfeeccjfihiahjdbfhdi
cbchecfjffj                             
#jfhdjbcffcbhhjifbfiahaiabijcffhichfehbf
hgghjjdjcbh                             
#cjaijcjaijddcfddjffgfaeehdcdhggehfcjgea
jbafegbhcij                             
#ghdfjdcicbafehijdafhfaigidchchcdgfjigfi
abiedbabhag                             
#ichicefchejjigbfdfafjdgdgfafhddgiabjceh
ibcficachhg                             
#ecbhgfadceibachjefafhgfbdebdeggfihjcgbd
ajjadcaeheg                             
#igigdgjfcjdgdaccfaiadjbdbifbheiffeidigg
ffgebidcaff                             
#adhibjcccbiibgacaddddjejfhfagagjjihahbc
fhfgbcgjijb                             
#cfiejhigahfaddiadiihiiajbhfagceghccejei
fgihjefeggj                             
#gbcacddebdajaefcgijjeahhhfjcdibchhcahdb
jgjhigafaje                             
#bbahcdcbbahacedehdjhjghdcaejgbjiejfaafe
ecghccfffhi                             
#jgaedeiagiaagdhifiibifbcbaihfedeiaihdaa

dhdecfbggib                             
#fchbichgiegfgfbceiddbffijfidbeegfdiefhi
icdegggbfhc                             
#dchjijiafcdfgifibgdjhdefbbeaabecbdffaid
gdbbdfihafc                             
#cgehiiabcjbceabiidefidiiheehedfgafddchi
djabbdeghch                             
#jdffbjjadhdihiegfhibhbifjfhehchajhfbddi
iaeacaghfec                             
#cdfahdbedfeieheeggiighjhefbddfhfjjjaega
ehbihicffgi                             
#jbadbaegfhjahceidagefjgedhhbfifgajahjhf
abdhhdacjbh                             
#ghebejcjehfddahcddjffhjahceahgicbacjfch
eiaggabdgjf                             
#eeegbdiccibbjjefebddgdceibiffebfidccdae
fibcbbjaaef                             
#difcijfcjcgjebhbhbffigajdgcabeiiahcdgjd
haheibbfhhg                             
#gjbbigiifbhhaehjjabieibiaaggggcebafgddb
hbchdaiicji                             
#bjgaiajdfagbdggdcgfbgeeibcgjdedbbdhjjfg
aagigbihhhb                             
#ggbcibbfifiddfeeeaecgejiaeceacfdiiehjbe
gdaaghjeecb                             

#agdcchjfddfdbcdbhdbjhhdfddagbchadhbgeed
dfhcdeccjjh                             
#eahaaeehdgfbddfafgibabidcjaaeaifjhahhjc
ieicjchbejc                             
#fgcbidcbacidgfhcjibidgjibeebibedajcajjh
ffhjdfeeifc                             
#fhdbejicgcfffbaegdgjiihgeeieifbjdhddbjd
jbhajgbeiie                             
#iggejcbhfgdehgeebdhajdeehdgghfjdjbfigad
dagcejbgbha                             
#cfcjigaifaefidbdbaghchhdhiaggedadbdfdhj
abcggcaaaci                             
#ajddigdghahjiageaggagcejbgjfgfggjejegbf
bceaeghggdc                             
#ebbicfhgacdiecjgbghafbcdbehhiegehaaebhf
bicjcjbcgie                             
#dcceehddbcgihcihgfcfhdhfceifjbhajhcbgii
ggahbjagbig                             
#bjgdedhcgdefgcedddjdffbihjcbdfegejedhgh
gcbcfhddajh                             
#baidjdihijiddacficccabjeecibbdichgjeihf
ighdcbjdgei                             
#feaidhehcjjahgjgaabadjibgheedbajacjgjie
hffjigfjdge                             
#behfcedbacibcdjhcidgghciddbajgfgacdceai

dijfhgeecfi                             
#jgabhcicdjieicdhfgdghehcgjhiiadcideghgg
ebgfbbaieih                             
#adjahadedigfejijcebfeeccehifefbcijhedfg
edbihadjhjg                             
#ddgabdcifejjeejhfjgeibjhbjfdhfehaefccha
ejiddeffbgj                             
#eebfhjifcjjbiaccdecbaecfhcgejdjhecfbbdf
chjehejibjb                             
#ahajefhjhcejaejejcejdfbjhdejdihahgaedjj
eajadhjhfff                             
#bgcbddfidbeafjdhegahhccddiddhjdegjdabed
ffdgiibidgc                             
#aaedgchgfeajedfdbiadegdchhhfcdjefdchbde
hgfhfgbfhga                             
#gdababjgeiicjgiigffdgahibaeiihfihgcjhfj
aechijgjjai                             
#cgedbjbeddbaiciiiibgiijegeigidhiahghadg
bfgjdcaebif                             
#acafieijigebehbajeaigheeabfgebbiifihcfd
edbebidfcdf                             
#bjcdifejgbgjfgcfaegebhecbfifbfgggibhcjh
gfcghjbaadi                             
#ebffdgebiagddeacffcjiheighccbbbgjahibgg
haacbgahbgb                             

#hcghjgdghbfhiihdbjcajfcgbghcacaeeagbgge
ifcijahaebh                             
#ihfefgefehgijfbjhibdhaefchajeffeejbhfbb
ccjbfdehedf                             
#jadcicjdbdfdbjhbigjbbehffbjadfbfccagjgg
jhecdfdjebe                             
#iafijhfchejeffbbgeehhiahbhhabdiefiiehja
jgbccacbdde                             
#hhbdajcgchdifgeabbbgebiecdifceicbcbcech
bjicgdchbji                             
#afgiiijbcgichdaedcgbhgeeibfjbjhhbjidggj
ccdgdeahcca                             
#aaaddgcaicabjfeffjchfdjcfjfeicihigijdhe
fgfccjeibbb                             
#ghhbhciddjebjhagabhabdbidhbhhccfeebddfe
haggdhbaabc                             
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fdhjgfjfebf                             
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cijfgcebahc                             
#eigjjehgjffhdggdbffifhidjjadfdccihehhfj
jdjghbbbbbh                             
#fedfccdigggbajdibefaihbdeecdgjhdcgbfhic
iebeiihfdfj                             
#ecedfjdeejfbdfdhffghbjjbdbdcfdidbgdhjab

hijfjfibahg                             
#ijficjjghdjacjigbffjegdgfbjhfbebgdfcgjb
hidccaiadah                             
#ejeddbfihidigbdgbjjgdfaijhfebjhgbjhajei
eigijcfcbie                             
#hajfahcggjifibbfbaadfhgfbibbeifbahhibhc
eafbdgcgecc                             
#ebedeacfhfhbddegjhiegdjbejeejhecebfjbfg
ccciehddabc                             
#aaegagfghiagjgifjefaeddgfcddjfgddddfagb
aifhchfcgcj                             
#hededjcafgedijighiifdhfideiiiibfghebjce
gcbebcdhdia                             
#iaaccdeigdhjcaiigafjhifagheeeigijabfbef
gadfeidgeei                             
#bahhigdebbbjdaidgcddibcjaiiehggicaideee
eeedbiijcef                             
#fgacjhhidcggggacfhffaiahheheihjigjdgheh
bceijbdjhah                             
#eefdiaidhachgbcfgijgacjccagidahjggfedja
afaahgeefhh                             
#haffjgdbggfcaiaahggbbfjjdchjcciidigciha
gacfaieadej                             
#biciddjjabaebgefgdiieabbcjhbchdfgahghgh
abcddfeiefc                             

#gifcdcjgaifahfebjacbdfeedijhcifcbacbffh
ggiifbiagid                             
#bibcfhjfefgjijfahaijhhffdajhaidgbbhjbgd
cabehdgbfge                             
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jgijjggidai                             
#fihfjfjbhiceihabbgecdfidifhbeaiddibffjj
iabgfdedigf                             
#ebaajcdaficibajeddhbhgcifjcdacajfbdjjag
achjabbdgib                             
#aggjeghbadchafbidiiejggcfjbdhhaebjicccj
adcdhgeaghb                             
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hbjdjcdeaeh                             
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egbjjgjebai                             
#hefhcgebejijijcaafhejhdcejihbdggjcejbdc
hjajfggeicd                             
#bdecedggiagdidadjjgacjhdacgdgfhdeggjibh
didjiifcgdh                             
#gjajbfbcaagbbdjcgaifcggbhhbfdgaagecajdj
ggjacijcdgb                             
#dbcaaeggedghebhfijadigjfeghdacgjhddcddc
jbfeiiffajd                             
#ehdbjhafebdcebgiifbbiejdjijiihdbbfhfgbc

fgbgbbefige                             
#biefjbcjdidiefedfagehacaieicjbfdbhiiied
ddfjjjcdidc                             
#faebejeehajbicaichihjcdbdbjadbccjifbegh
ibcafeeibgj                             
#acdaibdgabchicjbfeijdicdbagaceiffbigigj
cfdahfcjdjj                             
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gcgaafjaaaa                             
#dhgfffgfjadeccfdcegfibdahdfgfigeahahdif
faacdicbjdf                             
#cagcbhjhefegbfgjghificfibbcedigbjaeagia
cahhabifjbf                             
#ffffhahcbjfhdajaefgagdfbbebjeabcaceihga
cbjajajhdii                             
#hiifbaaihhaibaaibegjgifdhcdhiiabbjeijeh
ehfccjdcaaa                             
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hhghjcajjje                             
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cgbcbebfjce                             
#cjaiahjaifbgabacbdffjgcehabhbdcgfeebahe
aheciejehca                             
#ffjjcbbgfjciadefdcgcceaiehdfgcdfjaabeed
jdchbdhgffe                             

#aadcffacgjacfdghaedihbeehidgchcdiajdieh
haggjdbdhgf                             
#ahahicdfdhjejeahbfdchcjiafgahbhhgbjhbjd
bfbbggjbccc                             
#efjbcbbcdbgjceeiadacdaicbdbfdcdgeeggjic
acdedidjjhe                             
#aajcadebjhhjagfgeffdehehfjfdhggejgafgbb
bcacegdgcci                             
#dbgjccbgjgdifcedjajjhibbgfibdcfhgfcgiib
ifeibaibdga                             
#gjdeajacdbebhfgdedgiacehdfhggeajgjfehbb
edjgcgifghd                             
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dbabdgdgijf                             
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bfcfdgjihgd                             
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hefieifcheb                             
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dffijhhdgfg                             
#bgffhgachgaehafdebeafaeifedhjjceghecdff
ebadfiheibc                             
#gbhdbaacgbhdcbbgefceiahadjjgjfjdghbjgda
agcfhecjiff                             
#fjdaaiieacahhcghgigddfhjjiaecfaabijjeih

fcgcdgbbbfd                             
#ibgjdhccaededheihbgddgcibejjafdigaigeei
biabeghdeae                             
#aggcfbbifhdedeaffebebaighhihdchffgjbbig
ggcggbicahg                             
#hiefahhjeiacjafhgdfcjggiajihaciecedahbe
bahbbhdbjfg                             
#eecbciddbgaeiaidhfcjghdiggjcjjchaaijdhh
ffjafahffif                             
#cedcdgfbicjcdfcifijehiibfejcahfjhehihcg
ehjhfbegehf                             
#ibbdfhihdebijaeggfggiaaihchidbeejehbijg
agahjbfjafg                             
#adfihehfhdceehiahaehbbgbhdighidbicffjfg
dccdhdihaab                             
#eccahfadhafhgaaabgjcjjiabcgcagdfbfefadb
hjjcdfcgeif                             
#dgfcihjhdcaadagffaeeacdicjicajhaficjafi
bfdgigieicj                             
#ahggjhdaggciacajhichdiabgidheaaehbbiheh
dgcfcghacbb                             
#bahicgibdcfcaaehchhgchfjbebibifhbgfbacb
icbdfcfciad                             
#dcbaiadfijjaeggbjcfceghafjchcghdahihghf
bdcicfbhcaj                             

#icieaiijagbbdficcfagjeadjcicjceeiaehhbb
beeeaaaeddj                             
#gebbccdfacdhbebbbggfahcfchafejaahaeejai
dbbdgcefgab                             
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ajfbhahddcf                             
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dgjigidjahg                             
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bijjadghdfb                             
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dhbigaadfba                             
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adibafgbeba                             
#ddbcigfbjgbhhhbhbbbeiieahceedbgdbgadgbf
dfidbacbcaf                             
#ddiafjjibbgjdfbcfebgiageiicjhhicgcbhdic
iecfhjbiida                             
#ibajecdeifjieedaiaedacjjdhbgiafebjcdabj
chbaejgcbfe                             
#biffdeaeacafgfibbabhgfadeficaafgejgadhg
higechhffeg                             
#gahagdbdhadejjcdeebdhjceeihjecdjidfeacg
cefcfhhdfjf                             
#gaabaeabdjedgdggaagjfjjdgefgchifdbhghah

ggbidhabehd                             
#ihaicgjdcebehbiejgaggehiadcjeaiaagfbggg
hibhgdeafid                             
#bhhcigiaicfgchgiddfedhfbcagaehhfbeffjjd
fjfjcefefff                             
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jhdjhjhjjce                             
#bhjgjhiijjdhdbgbfajcagheadbaagihjgfdeji
fbfcbdbjgba                             
#aifhfgiiehagbebhgbfegjiacachahegjdbdhbi
ebjaajieghd                             
#checiebbehfaedhgjfbifcijccdbghhibedgbjf
eejchdchhid                             
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ibfcdhedcaa                             
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dffddchjcca                             
#ahiecjfheajcebegihjiefifiiffigghhhbdgdi
aegbgjceegb                             
#fahhfcegjgiefedfeebbdiieiagbcidbfhfheie
cdbdjjcebib                             
#bfbbhegbhfjaefjfjjjjgiibfbcjjcgaaeghggc
hahdjghbehg                             
#cbjbdhabcghdjdedhfihfcjcehjafceehbdadfi
iihcdhahiii                             

#dfiadbfcbceejaadjdgfjdiechidfbgdijceagi
ajdjghiicje                             
#jejbdbgacghaihhgfcbdjhdijdeccfcbebgfjfe
bfajbdidgja                             
#ecgjdgjegbcaigfgedaagichegigegfcdeagacc
cighjfheceh                             
#fidbfdadadjafcghecjgeabbifecacbbdffjjii
jagjhahedfa                             
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ibbfiggccec                             
#abjejefhdiagedffabefahigjjjaaagidciefgb
jgccehheijf                             
#ddcdfifaighjaabggeaaccdcciabjcedafddhjg
dcieiedaffa                             
#bhifeddbbcjidcifdejgffcafghiehfiedfhaih
cicicjjcgba                             
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cfijcegbafj                             
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jecbbbdbhjd                             
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hiigcibhjij                             
#igfdcefajhjegiahfafidaegcfiheedfajiiajh
gjdghjeajbj                             
#fieeggigbahgbeghigjdfifhgbfdjibjehccbhi

ggdhdfgddie                             
#jhecbhgacaeichedihgfcefddbddffdejjhdiac
cjbfgcbjjbe                             
#djbcfgfaggjbjiifhfebececihgcddeedgjegaj
dahjajidifb                             
#ieeeebchhbafigdebhgabaeiacjcjdbdbgeafbg
daihjaffjhg                             
#aahbigjhhehfjbaefibbejfgihbejhbjchdjbci
gebjahhchge                             
#ajggfcjffbaceajidgfbgbcbejfihdbafadffgg
fdgijbaejbj                             
#icdcfaafbcbjhgijcefbcchjcaagifieejeijcc
ibjcdbfgjei                             
#faicghjhedhhchigaaajhfjcdhcebfafhidjgjb
ggfchicjjac                             
#ihijiigedejbhefigjajfecfiedfagieeicafjd
fhdedbdadjd                             
#dghahhegefcfbchaacfcaagidbccegbicecjefh
jccifjiibfd                             
#iiceahbbfbgafbbhbjdfeecdjffcajjdaigeejc
fcaehfbcfgj                             
#ihefihhbgaaeaeacjdbefdedejebhfgagjcaihe
fdccfcacacj                             
#gfheddaeidabbgifccfdjgfbdhiidageajiaiah
hahegffdhhg                             

#fjcgeeegbdjhddbceeffdcgdffbbefhghffgdgj
dfhigagdgce                             
#bgifijhadbgcdjjdhcidfagieiidgfifebicdea
eafagiagfdf                             
#iedbfdhcgciadbjidibbeahiahabfahdaadiifi
gbefbebjedh                             
#cgchhjfgjiecbbacehcjbfaggbdejcgadijcdai
ccebbcchiac                             
#ibhjbcifficceehibfabijjdcjiacedjjdiaagb
egffeccjjdh                             
#dechggfjfbhcgefihjjhaefcegghagbhbbeibdf
egbfficieha                             
#hfijebfdaejhgdedhcedajbcbcjjdjjieffcafj
icjbejciahb                             
#iijehibdejdiihgeijcbccicbjffbihebicgiie
daidegjhdhh                             
#aiiaebfgabaabbddjcbjegdbabcfbifeacagabf
igajjeafbbe                             
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bdbdgjjhefc                             
#gcifiedicghdbbjhcgbeedidjeaijbhigbhifgj
aaifidhhcgd                             
#jfchaecehjajfhgjfiibehbejgdafcfccjeefec
bcghajjejda                             
#eeeiddcahceabhjjaaeccfeahbchajbjgecibbi

dhfehaecjai                             
#cbccicgiagejcbdgjgccecfeibdaghfafjeigca
bcgdbhgigjg                             
#bfdchhibajgeeahcejefjeeahecjgcefdhhajae
hfggcdcaecc                             
#cecjeiejbfejfjgdgfcfhhggihjhghfeidcbebd
fajbbicgagb                             
#jefabbjfigcjchbicjabgdfdabfhjbgbfefedea
ifhddjefejd                             
#hbdadbjaiefjfjfgeebaiifjejddejdiafibfhh
jfbeacgjcic                             
#bhajdjbabbdfdiiiegfdbifegehbjgbfgjfbafh
ejahgdfefii                             
#cgdecaaagbjacfcjbdbhcfdedjebfajdebhidbi
fjdehiacbcd                             
#higafajhagddfjfiaggcgefabdgieiiefheabad
bjdficedihi                             
#dacgjagcjfcdeijjeiegidhcicfgddejcegcaaa
jbiiahiebgd                             
#eghicjfdidbdidbbfijjbfcgcedhehiihhiceei
begajgbfhfa                             
#dfejjcfcehedibicbgifbichjbajddifccfjdii
fjjjhjjdjed                             
#fjjdicchibceebfhfgagfdbibaajjffjgjejffa
hfgebcijaad                             

#ahfhejceeedjbgfeadbefgfggbbgjhibadhhhhi
eecgajddhdj                             
#abdcdjfjefjdgfhdddedagbadddgajibicidecc
fedabbciaia                             
#dhhfbecafdfgjidcjfebaecghagdfgjedfijffj
dhhgjdbffgh                             
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ddhahifjbjc                             
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jccbcfeaigb                             
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feggeaijbdc                             
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fjbejbcfceb                             
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jdbifaajefd                             
#jahfbhcjghhcabahffbfifhfebidfiabfgdigfg
aeajfahihjh                             
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eebcihcgfif                             
#cbghcgdechjdajdjidihagdcddhacjfegdffdad
biihhhhibbb                             
#ddjejgjfeccifhbeeacfagchhjhjgfgedebiceb
eaaeeidjbbj                             
#aiejfedffafacighfbjeegdceaghfggegjjhfij

fjacheaghhb                             
#egdeaajcffbgibfehddefadgaaagaifgfadgeff
hfjjfjadgjh                             
#acgefhghibdgceiejaaecgcgabdcggiihgcdjah
eiajhhigiee                             
#dbjjaggebafcehicbghgihhdbgdgabigiihbgaa
dcbfgdcbbgb                             
#jfbabjdbiihaifcaibehhhdfghcgbadcgfghgjj
ddifhfjijde                             
#fcdafdchiccjijebdabehddggiabaeijidjifcf
igehebbjcaa                             
#eiabjhaafdeediebicfieieegagdiicfbddceah
fjhacdbhihg                             
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bdjgfcjhecg                             
#cafdhfebebbfcdbiggdffegejdjbeafheigchjf
dbbbccfajaj                             
#hcbgcgfcabhciabciaabifbaafbdhgfgdegejjd
cfhaefadaja                             
#adhdjbaigfgeihfgcdadhcedhaabaaeihffdfjf
dicjehidgjc                             
#cdajcbcggaheiacjgbiedgefafidggiegjeacjb
fdijbiiciec                             
#jgfgggifedcjbhhefagbdhdddfgeijajfjhceif
dffddcbhifg                             

#cbhjijhhbdgfbjhcbegbhgcghcffeehaahijiab
ihfdgdjiaeh                             
#bbcigejchjagfijjajgijegfcfheehcecjabhji
ghbjghjgccb                             
#fejaebhbdhhgeechghjahahgeajihjgfidifcfe
jjjeiecdjii                             
#dffgfebddfcdhfcbfgachghgjjiddhigbgggidj
aihechdgfgi                             
#fdgeifbdeghjfjdehafhcdafbiaidbjhibafgij
idefabahcaf                             
#dddaigdffhffhchiccbfhieaciddgdfggecfifh
hcjjhjcfigg                             
#aafjhbhhcjadbgejjebfaaehahicedbdiiiafej
ddedbiadcbh                             
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egdjajaehfb                             
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cijieccdjej                             
#bgdieiedeajjgficdbhbibaddjgdfhbgcchcchb
djgfdbigddf                             
#dbeghfgfhhhfdbchjdjiejcjdbafaighdagfgcd
icaieiejjfd                             
#ddbdfjdfdggbbhcdbgehdejadfdbjgbbbeghhag
gigijhcdigc                             
#bhbiihdibcfhiihjfhbdcjheejifgadbechaigh

gafghjehgbf                             
#gfhjbijjcchcaaiidiajehhdfihfhjbegigbhgc
ggedffdabhh                             
#diegibaifacddbfbjfccibifciegeebfbdhechi
hcgfjcjjddg                             
#acbiidcdjjehcahgciegedahdeiejajggidebgg
abafgbigjjc                             
#igdceahahfhgiihbfccjchgiggejjgcfideffji
bjcefjageia                             
#jiibabcfjjhcgbgibgbaajdebcaeihffbaceeib
jcchjdbgaba                             
#idecaachgffaijeieaiebahhhigfajacjajibaj
cchjegcebhf                             
#iajfffgiggeefjaghdgccbbfbicbdacaidcccjc
fechhfiddha                             
#ffjidgfaeifhdajfejfjdgdhdachdeejbfeeabe
jfciiidicdi                             
#gdifhbffaffddajffeeahjbacijcajehiejidbe
ificbicdhhh                             
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jbgaecddhce                             
#behcfijjgaadfgedghefccabbejibdejibcdfhd
gdfacjfefcb                             
#cheecdceihhefchfiiiffjaefgahfjehfabdgde
hhihhbegibe                             

#ihcjdidfccaccaacfbdbejdaaejcfgebffjeidc
igfdfeceibf                             
#decbcbgfccbbccaihfjgeajahedgfcdaicabiig
bhbaigbhabi                             
#cfiaggfecgcedjebcbabbbjegbcdfhcbdgejjab
iibjgjceajj                             
#ffegfbchdgbgdjbcfjchdfighheiiegbbebeijh
debbffjfbfi                             
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babbjegfdci                             
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jhefgeabhgd                             
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ecefjheeicg                             
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bbahaeifhij                             
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hfbhjdaabbf                             
#ghbeeijhdegbceaggidghffidgichcdgiaeggjd
ijfdjciabbi                             
#ceejccjgjdaabajbibdeafhceiajcceihfebjbc
bjiahhbiigb                             
#jbdjhdhddehbedefhdgcaeidhegfhbddfebheef
cciffjicjdg                             
#ihaddghchbcegdiiebcijfdcbififjfdjdbgeid

eiejgihaeje                             
#jcbbajidfighdbdcbhhjfheiaajhidahihicghi
faegeejhbje                             
#afgddcafdjgjeddcihjgehaaficacfdjbfiabia
gafdbiiafcj                             
#cheifecafcjjhbbhjfgjfididiaaefjbffaebab
gchehdeagcf                             
#idcdjeidhdaddggeijcfbjhbbeefdgicdaeebga
idafhfhjjdi                             
#bahcfiieeeaghadhedggdhcigebjecjdbjbghhf
eejfadiecdh                             
#dhajegbghjcjjdadafihcgdijeifhfafdjagicd
accfficcdjg                             
#cgcdebgaghdddiabidghdchddcajdfbicegciic
ijdjijbihja                             
#dhiihfdidicdhdgjdafcicjgjbgbbdgdfebbjjg
ifcabafegef                             
#jddfbbfhgceifcbdiddhgafdbjiaiigfhjijfac
gaahheidaeb                             
#adcfffjeebfiheiafdfbfceccbdddeeiiffbfbg
bhdjhjbeeib                             
#dbjedbegehicdcihfdjddebhcfjhachiighjjbe
iaccjcjaicj                             
#icehbbeagefcdfgijdhcedeajfhechfjdhfabhf
ficaaffjbeh                             

#hhdcbhcjjciheefihejheefddaccbfbedbcafcf
biacbcfbhed                             
#gaaffggiijeihdejaibfidjgcefagdhhbfiiabe
heahfghbidj                             
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egiehhjhjfi                             
#igbbgfiibefjbjcgdfbhhbbdifghaahhagfjhgf
defdiacghag                             
#ahabahijbbdgageidbeefijaicejbgiebcbjhfi
iihfbieeeif                             
#egiijbgiaddcdbcfgehgfffbbihcaddgddeggfj
eihbgcahcbf                             
#ecicbgagedhdbdjgeeadjfaeejhiabjdhdbgeic
jcdegcbfahc                             
#gdfheaiaeijfhdeafdageddadadfchjcfjidaci
gifhechhcde                             
#jdafhehibdfdabgafhfcbcddfhbbehiahefhfbi
dedfjbbjdib                             
#jefeiaggigacacehbheaedbjjddaifdfceccacd
dbjegcfcdca                             
#jgacjhdcaeeaaicjbiejgggcehjgfgheccchchh
jfjdcaifggg                             
#aieejdhgfacacdihijijjaiahcibhecdghjfdbj
ghafaihefjh                             
#ifggdgbfjadciagdiifdgjcceggdaaeecdadbca

fabefbjhfih                             
#cceaddaeddcbeefffaahbjfgfedfcghbgbbebjg
gcjbebadjjj                             
#chajhfccjeidcibgfjfddifaejajceeaeaiefhb
idjgifhebfa                             
#eaaiaijigcaadcadiagfcedfddffifcdcbbijhi
hbjibcadgfi                             
#caiafhidjchjeabcchgfejddbafggfbbeeijihe
jfhddhgjjhe                             
#ggchaifhaiciegdeigiciaffhiaahggebjjcige
dhdhidgeche                             
#cihdgcaecijbfiggajaghfefgbjijbhjfgiifbg
dicbjgbadad                             
#jbigchadehjgejdjiiefafeijffaedddhcffech
eicgcdjcagb                             
#cacdhaccfeedabcjhfggcjfcahbaffgdeifihdb
dcadggfcdad                             
#jgadjicbgdjaidhbfijefdhfdijicedghcjiaif
fhhahdfhidd                             
#ighfafafhcgfbdaahhjjdiieieegfcfeccibbjh
cbidbbfciji                             
#dijedcdbabjbdbcjbdbfiabhaebjbiigjihaife
ajibaeddhhh                             
#ihahjffijjjccfjgcgadgajiadhdafbdjhcbcid
ijbabfgcdab                             

#hefedfehjbgbfjibafcdhbhjhehhjgdhheejabe
cfdggjbdiba                             
#gdedahcigabdahgabdajhdbaefiffieigehigdh
bfeacecejdd                             
#cgihbjcjhcifegeiiigeedhadcfejdahjecfjea
cabjhfahjjd                             
#jfcdhabaacaahjbijhhefdejgcdhehahhebiiab
hccifadgebe                             
#bafdjbafdagieehdhfaidghhdaddaeghcgdcjfg
gdhgfjaaidb                             
#jhdfiihdfibagfffacjffjigjheddfjcghihjif
bgjehibdgid                             
#adfacfjgjjaceadjfjjjjihjfgeebicgciebbej
iijhfhcghjf                             
#fagfdhhgaadadhifhejicheajadigcdfbijfdfc
ihjfchgceec                             
#fhchjbdhbbifaigjiecajgfghabeiaecjcgjaeh
bbdajbacbce                             
#eddfejcifecdabcaafhheegajcbibgdfegicadj
dedhihejghj                             
#gfjeabhifbagieaabjjcbfefgifefbcdhchebjj
gjjiidebcgi                             
#ahfiegbiagafjiibjbchdbfhcfjccajhggcechc
dcbdbighhfg                             
#jgfhdfcjdjccbeidjhdgiffajagifjacfdeddad

eegdieefjfe                             
#cfgfaighbefihiibhidfbbfchidbgdeidfajjba
deajajfbced                             
#ggddefghfgedhhdbdgjigfidccafeffjcfeidgh
eacajfbhdcb                             
#jhaababeaaafdgedeigjgeieacagjfggaibdiii
bhjhegbddah                             
#fghfgcjefdeeiaajdiaijfigefjbfgjjigjgidc
ibgjfbdfdgh                             
#ciacggbehfbdfddeedigigdjbgejeaaehafibbi
edaffgahhfa                             
#idhcagjjcbhbagedejcchibagchgafceiadicjh
fbehffiejdg                             
#jdhjigbffjdghafadcbhgbejagdgbibfhhcfifc
jeiabbgfeie                             
#dfibgbeafccidbcagjdagijfehgbcgeebhbhjfj
jaadiadadda                             
#djcbedgcebeaecefihcdiifhcgejjdbdiabehca
ihfagcjehag                             
#iaddfcdjhfdcfdfeagjadchdejefjjadfaihghi
eggijjcdaje                             
#hiiciehiaadgccaadbggcdjaahafbgfiiafjgce
ffehjfhhcge                             
#jbhjahcefgaaedahgdahebgdjbedggbifggchfj
idcggabgdbi                             

#aeehbcifhdbabedajcahidcciddfiiehhgjeifb
bhihdcaihbb                             
#fhhcgbacaihgjghfgbaefbjdbdaijdhefjgiice
hbihjdbijjb                             
#jhfjfdegicegiaebeigjabecgcbcajecafcgajf
ajibfefcaff                             
#afhejcehgegaccdfhjfdjcjgfgiidjdddjfjghi
ghdedejiiae                             
#jehgdihfichaccehjihibeegdaeegfffbbafhae
iddeiijbfjd                             
#gggfeficcicicabafedhbdighccagaccdajiehd
ighiahgdbed                             
#hgghcjeabedegbagdejfegbdcdceeahhaeidchg
gechedfaeah                             
#bdbibcheijbfhigcdaabbiibifhddaaifaifecb
jdfhcehhegf                             
#cdfhggegghjeediggihchigfeiaidihbicadifg
giafhddchdb                             
#fcdcfbacfihecfbhjbeigchegjcdiaijieefjch
bcaciiagage                             
#giajefgjajhhbjbfiahdghejieebcehebgegeib
fhidaaghigd                             
#achjiebfhiadjfbibaagbcciigjddaafafbdbga
ibjcjigfcaf                             
#adffdeehddhiefjceccgebabbgihgecgbhbicda

gjbdbjgjfij                             
#ejeibcbfagijhfdbfeicbbhcgijejabheddgcca
ggehcaecdch                             
#dhacbjiaaagdaidfcdabachdcbbhfeacdjagicd
cjadjgbgaef                             
#hahhgbgcbbhdbeecfdbbfihhfadfifeaicecafj
fdhbcaaiafh                             
#iaegdhjbcjgcedfgdfjajacaadeigbcgjfcdfee
adaijegebcj                             
#hahjgaecfffjedgghajcaahbcjedcghbdcdgeef
bfihfebfjih                             
#ibijiddhcbgedacedeccbgcjibahfffgcjeiibj
ggjbaahgcge                             
#djcjbajcabdgabejbcadceijdafgdggcigibicd
gafifigejcc                             
#eegbjfachjbaajahibafdeabhbdgfjgejedabcg
gchcagadhea                             
#fcddfcfiifdggdjjchcejbghhhddjdaeeddgfjd
cbecbjgehec                             
#fbbejejebgejhhjfgfacdeidjibfgdjfigejbjd
gicbeebhhig                             
#jaaacjcejhbfhjieheafjjjdagebcjbhbjcegdh
heibidicbec                             
#hicjcfaaahdhfjdefajjidggcegagjdgjhjjjdj
ijafgbadejj                             

#gddjebgjcehgbifjhdeejgbgbaddfdheehaeacd
ihghjhjfjhb                             
#acebabigicgcbgjddccbbeijajfbijdijffecgj
cifffahjfgb                             
#ghbdjfefbjcjdhiighfcgghcabbcjdbhbjiijah
jhdjcgified                             
#cggahgdahdjbagafaidehgjddfdjajbbfbgccei
ihahaeheicc                             
#fedebcbfhggeihahaibgejcigichegjieafbbab
fajgccejdbc                             
#ggfihidfhcgbagahgeghhjdajeebfjjdhdidfja
fdibjcededd                             
#jhgdfbgechebjgcbbicbgjcgjcdhjhfgeghcfce
dgcacahjfcc                             
#djghhabhfjbfidheajaihfijaedbggjbcbdjifh
aeijihedfef                             
#fahcffadicbiahibcbhjecabhigdedjdiejgiih
bhcjffdfajb                             
#gbfidfghcbcfiaaaddcjhbefdahdeagbacdjeij
idfgidfhdgc                             
#gcjgdjdfaiegjgghgfgabjgbdfhgjaiaajghhhf
cjjeeedgfdb                             
#jhahhjggahaaifahdaaihccifhegdbebfihhcaf
eddebheafdd                             
#cgjfdgdcfjaebgicdibdcefdicbegfdijceiegb

dchdeejfeii                             
#ifjibdaabhdcbgbchigagihhbfhbedbjbejdegb
bdjfcihaibc                             
#jjieefchcajchhbcfedihjgdhiahjhbffbdffie
gjhejdcjgfg                             
#djjagghddfedeiaaiccdbgiiachieigbifgajgj
cafadjehied                             
#agijfehcgcgicieaidigcjcbfccjecdiaiijhib
gfgcaiebdja                             
#cjbdchgfgdeeafceiabhdbffaehhgegfccgcjjh
cdggdjbfice                             
#hdjfbidhfjabhdaacjebjfecbhcciggicehicfa
jghaahfceea                             
#deigefibfbafgaeejfhdicfbibcbgifddgcihdj
fcdegcaddge                             
#bcahbcgjheefacfheafebhecgdgiefcdhggfaea
badbcjieheb                             
#cgejchajbejcjiebgdbaegcbbjfadfbjhciecjh
ejcbadjfhie                             
#gbcgdhddhgiaaadahageiacdegigcdhhfgfehfd
ahejihfcabh                             
#ccigggaidgiaijbgdeaedabjbejiibebehgehae
giiffhbgcdi                             
#fhddhafcefbbdhfhjbhgghgcffjedagcdafiiia
fcgfeffeche                             

#hajiefcdiaifdbbdfdajjbafagdecjcfafffdfg
fgghddicaia                             
#fhgfhhjcgfjdcdbdeefhbcbedjfcbehhhhijeaj
beeabehgbch                             
#gjbjjidejgdeiifefhbbjgfhbjbbcgbdjabdjbj
hbjidbhacfh                             
#cafbjgaajdbffegbheebcdcgeebjciideceejfh
gjdihehbaea                             
#hhadddgihidffhagfceijdcegbdgfgdhgchgjca
cachfgbgcff                             
#fdbcgfiegbfbjcdfacjeagfbjhfifabbgjhcdce
fbgabiccbbg                             
#edbddhadddaecigfbgbjcbhfjcbcdacgchcdaah
cbbhaghggah                             
#ieijgefabjdgddfhfjibhhddcffejgjfecbfdja
djeicgdcaef                             
#agjbabejabaebhighbidehdfciaciaiebafgabe
edaeiaabjcf                             
#fbfagddfihhihigchieceehigfabijificefdca
jagfjeecdac                             
#jigfafbjjgghcagffcgichciagbiiahbbbaggga
fiiheeebgfc                             
#ahgfdaecfbaeijejbieffjjfedjefjidbiabffg
ciajehcbehg                             
#egghiigeiahejgffcbfbffaicgaibeifeeebigi

aeeadjefcai                             
#acbjifdeijcibfbdbaddidiagieihfgfbgdahge
agjfefefdde                             
#bfheaiehhabifedhcdbhhbjcefdhdjiiiabgdfb
adehaadihgf                             
#eidjhgddfebdgaechihfeifdhgbjcbaajjgicie
hafhdaghfec                             
#hhgfaggejcjdbdacbeefhahcgfdcfbdcjhiijbb
eajjehiahff                             
#cbebhfhjdjihigfiebfeaebjgfedaichjefdbif
agggejcghch                             
#dbdggjcdafigdecegadafhhgeddfffadgffhejh
dgbgifdhdeh                             
#dfegjfbbgggdccdaeaaaibiigeababjdccajdje
dibbihjfbdd                             
#cjfaigjghjabdhhehccifdjajhddbjhiedigbfg
hecffjccbhc                             
#dhcgjhbdhdfcccfdbbeabdbjcdbeedcheedgdgi
bcdijedbdhg                             
#cfigcbdabaeejfbiggfiejdiegfidjcbbeihhdj
ebgcbchcchc                             
#hgjjggcedgdidbjffbebibcheefhgedgeiebhcf
gcfahifieje                             
#jcbedigeajedgefheehbijdbfdbhjbhhbbcacce
hcjagaacdeh                             

#faigejaighddbjbcbjdaghegefjdgajehheijie
gfbcdfggjef                             
#ejdcjgghggehecehgaedhiichejjheeahhieaaf
gjjeeijfcad                             
#fdifchdfdefbbdbbdaheaiiheacajfabbcjfgfa
giijdhiddbh                             
#dhhcaghbeagejfeddcdfechdbeaccbgacjjbcjb
djdchcjfbdg                             
#bfbaihcjfajdgfjahehijgdidchabaejigjhcgf
diibidhifhd                             
#hcabcighigebdjjeiabjcdcgjfifecjffcagcii
bcdgjfafhai                             
#fhagbjbfcijjdfbcbhjheecdjgfjacifdbibcfd
dbjabcjdgfg                             
#hbbfgjchccfbcbehhebedfgfbfhecfcfghbfaej
gecdjecaeic                             
#cgcacjgadeghihiajhefibfifjfcgcdeigdbhhd
jbbbejjebij                             
#aeebjiihhhdjdbhjicdeciagbahdgehjiagjcge
acjfjhacfdb                             
#jihdheegbijgffiehbhdbaheegcfjiiigaigfdj
cedjdigjgeg                             
#jcddehbjaejaehbegjbhhhaacddebjdddifdbee
bdajajcabdi                             
#fgffbfceebicahadcdidicgicbfjajdhbggbdcd

ahdgjfciidb                             
#hiecddfbbbbjjdabiicdeadbbjgddhadaijcccg
hicaeeejfhe                             
#bbgefcjifhdeddcfciabahgfgjjhafabhdggdba
bccchiihhcf                             
#adhabchbjgijdbfcebfgbjbafadbgdhejfbbfah
hgehcfidhda                             
#iaagbdgidhccjjjdaehbhgcdjjeabcdjahidibj
ijbhgfgdagh                             
#feddjbaabefcfjigejbjgfiaejafajcagcjfheb
deaiacjhfid                             
#iaeidefiacibgifbdeibhajhiiedjgfbehiiedf
cedhebbched                             
#dhgegiidadgjicjfejggbfgfhdeabejehhgcdii
eghagfhijee                             
#bhbbefgdgheiefbjbagehecfbjbdaehjgcchige
cfbeidcefic                             
#aicfbgeadbjbcejgebabafifaiacfgbabbcgice
dhcdcbahige                             
#gebbaicgdieaaeffjihcfbgdcfgafgdfhhgfjci
igdefhgfbfg                             
#ghjbejhbghdefgedjhceehadegcejfgejficaja
fdaeecjahfa                             
#gedgicffbggbidjhcdibbbcdhgahicggcgaaicc
dcefdgchcad                             

#dgbahjidaacechchidcdbhhcdfijedidhdgbaef
gdidhigaffc                             
#egdahhecdihihfhggabecgbhidifgbagjaecgdi
fgbjdjbhibj                             
----------------------------------------
-------------------------------------   
,champernowne constant1 the natural     
integers concatenated4 this is a        
,,normal number in base #aj1 ,ref3      
,d4,g4 ,champernowne1 ,the ,construction
 of decimals normal in the scale #aj1   
,journal of the ,london  ,math4 ,soc1   
#h1 7#aicc74                            
#j4abcdefghiajaaabacadaeafagahaibjbabbbc
bdbebfbgbhbicjcacbcccdcecfcgchcidjdadbdc
d                                       
#ddedfdgdhdiejeaebecedeeefegeheifjfafbfc
fdfefffgfhfigjgagbgcgdgegfggghgihjhahbh 
#chdhehfhghhhiijiaibicidieifigihiiajjaja
ajbajcajdajeajfajgajhajiaajaaaaabaacaad 
#aaeaafaagaahaaiabjabaabbabcabdabeabfabg
abhabiacjacaacbaccacdaceacfacgachaciadj 
#adaadbadcaddadeadfadgadhadiaejaeaaebaec
aedaeeaefaegaehaeiafjafaafbafcafdafeaff 
#afgafhafiagjagaagbagcagdageagfaggaghagi

ahjahaahbahcahdaheahfahgahhahiaijaiaaib 
#aicaidaieaifaigaihaiibjjbjabjbbjcbjdbje
bjfbjgbjhbjibajbaababbacbadbaebafbagbah 
#baibbjbbabbbbbcbbdbbebbfbbgbbhbbibcjbca
bcbbccbcdbcebcfbcgbchbcibdjbdabdbbdcbdd 
#bdebdfbdgbdhbdibejbeabebbecbedbeebefbeg
behbeibfjbfabfbbfcbfdbfebffbfgbfhbfibgj 
#bgabgbbgcbgdbgebgfbggbghbgibhjbhabhbbhc
bhdbhebhfbhgbhhbhibijbiabibbicbidbiebif 
#bigbihbiicjjcjacjbcjccjdcjecjfcjgcjhcji
cajcaacabcaccadcaecafcagcahcaicbjcbacbb 
#cbccbdcbecbfcbgcbhcbiccjccaccbcccccdcce
ccfccgcchccicdjcdacdbcdccddcdecdfcdgcdh 
#cdicejceacebceccedceecefcegcehceicfjcfa
cfbcfccfdcfecffcfgcfhcficgjcgacgbcgccgd 
#cgecgfcggcghcgichjchachbchcchdchechfchg
chhchicijciacibciccidciecifcigcihciidjj 
#djadjbdjcdjddjedjfdjgdjhdjidajdaadabdac
daddaedafdagdahdaidbjdbadbbdbcdbddbedbf 
#dbgdbhdbidcjdcadcbdccdcddcedcfdcgdchdci
ddjddaddbddcdddddeddfddgddhddidejdeadeb 
#decdeddeedefdegdehdeidfjdfadfbdfcdfddfe
dffdfgdfhdfidgjdgadgbdgcdgddgedgfdggdgh 
#dgidhjdhadhbdhcdhddhedhfdhgdhhdhidijdia
dibdicdiddiedifdigdihdiiejjejaejbejcejd 

#ejeejfejgejhejieajeaaeabeaceadeaeeafeag
eaheaiebjebaebbebcebdebeebfebgebhebiecj 
#ecaecbeccecdeceecfecgecheciedjedaedbedc
eddedeedfedgedhedieejeeaeebeeceedeeeeef 
#eegeeheeiefjefaefbefcefdefeeffefgefhefi
egjegaegbegcegdegeegfeggeghegiehjehaehb 
#ehcehdeheehfehgehhehieijeiaeibeiceideie
eifeigeiheiifjjfjafjbfjcfjdfjefjffjgfjh 
#fjifajfaafabfacfadfaefaffagfahfaifbjfba
fbbfbcfbdfbefbffbgfbhfbifcjfcafcbfccfcd 
#fcefcffcgfchfcifdjfdafdbfdcfddfdefdffdg
fdhfdifejfeafebfecfedfeefeffegfehfeiffj 
#ffaffbffcffdffefffffgffhffifgjfgafgbfgc
fgdfgefgffggfghfgifhjfhafhbfhcfhdfhefhf 
#fhgfhhfhifijfiafibficfidfiefiffigfihfii
gjjgjagjbgjcgjdgjegjfgjggjhgjigajgaagab 
#gacgadgaegafgaggahgaigbjgbagbbgbcgbdgbe
gbfgbggbhgbigcjgcagcbgccgcdgcegcfgcggch 
#gcigdjgdagdbgdcgddgdegdfgdggdhgdigejgea
gebgecgedgeegefgeggehgeigfjgfagfbgfcgfd 
#gfegffgfggfhgfiggjggaggbggcggdggeggfggg
gghggighjghaghbghcghdgheghfghgghhghigij 
#giagibgicgidgiegifgiggihgiihjjhjahjbhjc
hjdhjehjfhjghjhhjihajhaahabhachadhaehaf 
#haghahhaihbjhbahbbhbchbdhbehbfhbghbhhbi

hcjhcahcbhcchcdhcehcfhcghchhcihdjhdahdb 
#hdchddhdehdfhdghdhhdihejheahebhechedhee
hefheghehheihfjhfahfbhfchfdhfehffhfghfh 
#hfihgjhgahgbhgchgdhgehgfhgghghhgihhjhha
hhbhhchhdhhehhfhhghhhhhihijhiahibhichid 
#hiehifhighihhiiijjijaijbijcijdijeijfijg
ijhijiiajiaaiabiaciadiaeiafiagiahiaiibj 
#ibaibbibcibdibeibfibgibhibiicjicaicbicc
icdiceicficgichiciidjidaidbidciddideidf 
#idgidhidiiejieaiebieciedieeiefiegiehiei
ifjifaifbifcifdifeiffifgifhifiigjigaigb 
#igcigdigeigfiggighigiihjihaihbihcihdihe
ihfihgihhihiiijiiaiibiiciidiieiifiigiih 
#iii                                    
----------------------------------------
-------------------------------------   
,copeland-,erdos constant1 the primes   
concatenated4 ,proved to be normal in   
base #aj4                               
#j4bcegaaacagaibcbicacgdadcdgeceifafggag
cgihchiigajaajcajgajiaacabg             
#acaacgaciadiaeaaegafcafgagcagiahaaiaaic
aigaiibaabbcbbgbbibccbcib               
#dabeabegbfcbfibgabggbhabhcbiccjgcaacacc
agccaccgcdgcdicecceicfgcg               

#ccgichcchicigdjadjidaidbadcadccdciddcdd
idegdfadfcdfgdgidhgdiadii               
#ejcejiebaebcedaedgeegefcefiegaeggehgeic
eiifjafjgfacfagfaifcafdaf               
#dcfdgfecfeiffafgcfggfhcfiagjagjigaigbgg
ccgcigdcgeageggfagfiggcgh               
#ggighjihaahbahbchbghbihcihechegheihfchg
ghhahhchhgijgiaaiaiibiicg               
#idaidgiecifgigaiggihciiaiigajjiajacajai
ajbaajcaajccajciajdiajeaa               
#jfaajfcajfiajhgajiaajicajigaajcaajiaaag
aabcaabiaaeaaaecaafcaagaa               
#ahaaahgaaicabjaabacabagabbcabbiabcaabcg
abdiabeiabggabgiabhcabhia               
#biaabigacjaacjcacjgacaiacbaacbgacfaacfg
acgcachaaciiadjiadbcadbga               
#dbiadccadciaddgadeaadecadeiadgaadhaadhc
adhgadhiadicadiiaeaaaebca               
#ecaaedcaediaeecaeeiaefgaegaaegiaehcaeig
afjaafjgafjiafacafaiafbaa               
#fbgafcgafegaffcaffgaffiaficafigafiiagji
agbaagbcagccagdaagdgageca               
#geiagggaghcaghgaghiahjaahaaahbcahcaahdg
ahfaahfgahgaahgcahggahgia               
#hhiaijaaijgaiacaicaaiccaidiaieaaigcaigi

aihgaiicaiigaiiibjjcbjaab               
----------------------------------------
-------------------------------------   
cos7#a7 to #aejjj digits4               
#j4edjcjbcjehfhacigagdjjicffjgddbigffjcg
cbcajdbjfagibbbbgfgjjigbeechaajjcidggddg
agfdeagieahefjhgahcjhic                 
#dcegagcaafjjcjjhijighfjfccgfjjbaffcdefd
jfeabbfedagcahehdgagigaafddgddgididbccaa
gibdeeacicbedcceidcea                   
#ggefgjbhibeifcgegcfaedcbgedifdageddiagg
eaaeacabbbgcjajjfcacegjghbcbbcfggadjaeag
dfhiieicffghgcjfgdbbg                   
#fbjbdejggfcgddjfgehgdihafaghdbgbjbafdee
heaaaefcbifhhijegajhabdbgbiccafihfhebdga
defhidijdcdbcgedccjid                   
#dbcjbdjiceifbciehcahbdedgbhagcffdjghjga
bdcdccfbagdhajjcbbjbgabigeghhbbiagfddfhc
eihgbfiidbfdiacddciah                   
#bfefideceaegejgfbghbeachjdiiafjgcjfchjc
agbaddejcdihfabidhhccfccefeeggiijigicjae
bhgibghhdjchihjjigded                   
#hbeajdiibdecgihggdjjfadecggfcgachghccei
dbcdebdafhafdbhcfahhbhdhbcgdhifcbgcijeef
bfjiabjagehihbgejbebh                   

#eiiiagdchehjfibdheehdbcbbaghbfhehbgajhh
biaefdfhcjjfgifhgeieeacjjcfajhabjccfgdgd
gbgdiahajccfgceaejicd                   
#ehhhhcjdbjcbageifeidjebgjcicdgfbejbdhgc
gjgebffacaccfihdbdafjeigajeiefjfeiiighfi
achddaeegddaddffdbjja                   
#bhcicihhgjibfcbcdeccchhfhfbfbiifedgjigf
hjedhcfhcjcehbaahbcdaagcbdahdfeggahfdaaf
eadbidahhcbfdddfijghc                   
#heiacbaajhifegeajcifjgjeifjgeibbaccbcff
jagghfceadibajhbjejcajfefedaihgidhdecdbj
cfegechdjbihgiefcafaf                   
#haegjaaihaigjceehhgedghaebggdaiifbhficf
caajbfhfhhhaahicaiafhbiiggdhjagcahgjceag
hffihjgjfdgddehcddjjj                   
#cgfhfjafiijfbcjdhficieijgidjfgjbfdjfefg
cdfbjaahcdicbgafcfeiijfahcacghidfbadicde
cfdbigacibfbjacajbiej                   
#cdhdafbehbjeahjiicjeiddfbajggdihghjbhhj
adehjhifaeidcedhieajbhbgdgifdejfjaegbeeg
ejicgfieccchjbeajbccf                   
#ejcjachchbebbdccheaeeaibbbbhacdegbhcibg
aacdjceibeagcddacijijfegfahfffaffafcggii
ibfhiehdifaajjejhieid                   
#eadjiajaaaijeacbehjbeiejaeddcjieacgbcih

egbjicidgfdgdhidbhhjcdaceffbgegdegddcchh
cdabcihgeefgbegihgghd                   
#eddccihjfcefehhfajfccajheceegdahagcagef
ijjachgaifaccdefiegeihjgagfecdafbeegcidc
ffgjcfbbaeeebidjcdeih                   
#egjheebhajgcheacjcfbbdbidijidefgafbcebc
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giehifiibdjjgddajceae                   
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gghfbcgfbaiebgeaigeei                   
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ijdfbajcfaaiegfadadbi                   
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cehhbgjfiedhhbfbfchbc                   
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jjbihccdhhaiaeagcehfa                   
#deihgcdjhffibdcddicfggcbiehggcfeggaaibd
fafchdjjeedgaabihdgdfbhibdjghcgabhfejchj

deicgfaicifgeedaejehj                   
#ebdbcbhjaijhiachfifcgdidhdgfjceegeghcbb
hbjdjafbghdigjahjdjjfiijcgbdijcifgiehdhd
ceifbgidadgieghdaehgg                   
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adeiibaehidgfchcajihj                   
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jjigbfjhhfacbgbjagjab                   
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gbcefhgejibcfjecibiga                   
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ffdaibibfcdihfichbjaa                   
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afdjigjdhcjcajfgjhibb                   
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ajichjhfeghfhegbicdfj                   
#fddfbfdjadcagggegcaabeeaeidfgadecchjgfc
ibbahbehcfbjhjfebdgjigcigaijcgiaicdjaggh
dfefbecfehjbadhdgabib                   

#ebdeheeacgbcdjeeifcbgbghbfheeicbbchghga
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eiecaabeabaejbbabcdge                   
#jefeghdeidgjcedbffdffhjabedjhaafdbjaeie
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dfhfcbbhjiebfegjeeibi                   
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igihbebjejiaedhdjhibe                   
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hcjgifiediacjgjiggjjc                   
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bacgihgigijiiiiabadhh                   
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ahcijeejadjhgjhcbchgf                   
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iacdeahddccdddccjbefh                   
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hfaedjfbccbfchhabfjadhibfgiiahaicfhhiehj
gdbcfgdagfdhcgbjfjieg                   
#hbgcjieagbhfiehbaibjjcajdiggefabgbhgede

hfdhabiffhddcjicbecchbcicfhfigdbfehdgcii
icdfgdiigjhachhfifejh                   
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aaciedcaigeifehcdafig                   
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bedcdeddjhhibgjheeaeg                   
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fcdjfbibfgahaadaiafbh                   
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eeagghihbcgedeeafcbha                   
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bigjbjbcjbjdgchhbedhd                   
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ebfgfbhadahfahjaeideg                   
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cieedheeacjjgjibegead                   
#ghbjijeciegcegbbjeeijdhggbifffajjjjdbii
feghgggccdfibieefagcajaeeefifgdahehejjjh

fffibfifajdhicjbjfafg                   
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cijgghabjhahighbhbgga                   
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djjedbfjdedacibdhaafcecdfhdhehjdddfcbiib
fiijdcdeahjfjhcibjafd                   
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eabahabacjijbcaeijfff                   
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hjjieajchhgfbgcahbgjaciajceejfdfejiagicf
gcdeeciaajeijheihgahc                   
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ejhifaahajghfbhdfjagd                   
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fhfffijiehiebbbjcabib                   
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aaafgdcfjjidaacdedddi                   
#fefajibbifijhcgfiahfagbeaabaaddafcbbcha
jecfdafidfahbbjjfdhfgjjgidjafacacgbijiaj
ijefcaaabdbijibfbiach                   

#gdficegbfeicbeggaccafhfibgdabaechhbgdie
aebhbcfgabgabihebbigfbjihgbdeagghgicdcca
gjdjicjbffbgedjhhefid                   
#ifceggbdgjbhccdbeghfbdhjfhadadedjighadf
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ffagfdegihhdcffbhcfhi                   
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gdcafidbgcdcbbahgihae                   
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afefahhgceghdbefaegcd                   
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jacbghhjjcfhfbajgdbha                   
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hgdihejbfjjffihhcghfj                   
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adjaibbfbhabfacggifhahaccbeffhgddccijdif
cdgihhdcchidjiccfecai                   
#ghdjfeaafehajijfbigfaaefcfiefjgfcfcegda

iheaibjgffbjijjdiacedcfadcgdjcabedccibij
fhicfeihihidijfbdcfie                   
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dibeahdefcecaahhebhjd                   
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hhfiijhfjdiggciajjdbh                   
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hiaeidecfbebacijbbihg                   
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hjhjiiabhbbffiejgcigi                   
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aiibdahcefdhiaebdihjg                   
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fjfgafafjeadgcedgbhjb                   
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jceabgceijhdebehfajee                   
#affcfbeifijgbhabahiffehaabedjbceifibadf
iaceibbcefaaicjeifdiigfddajaieefcaibhgih

caccibdeahccddajhjgbi                   
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bcfajgjfgifjhecchbegbedajaidgbcacihccjfb
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cdjfbghgcgjbhfchbbheg                   
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ijaebcadhhcfifgigeibj                   
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iiajchechbcbcbfcggbij                   
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fegfbiigcciiiadgjibeh                   
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hhhdbeicacafebeigcchf                   
#accjabecfabjdgicfhibibebfbahiiiiafgiech
hbdaefficchdiadhbbajhaigcedbedhbddjieadf
afcibdgfbjdbaceifadjc                   

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dhcaicighbcjieajihgaggaehfiecfihbdjcdadg
aeicedjjiebhfiicjdbhf                   
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ggbdedchibhgiiehhhcje                   
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bccgffffafjbbhfebeead                   
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hiegbhdaijbjhahfjaajb                   
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ahfedadfaafiefgiibdaahjbagdddcfafjahjaba
djhhajihgbchchfgecabg                   
#geahedcgacdhjiihbbgcdhagebeeiaedachjecc
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dechedfbiafidbcdidifi                   
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bchheidaaacbifieffeaa                   
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jehgjfdfefccbegifeeib                   
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fgifhdgbeafjacbfcjcje                   
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abfifgdhbaahbhjdgacdi                   
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hhchaifhgidaehfjgfjjb                   
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ggbeiecdhahabhhcgahaa                   
#abacecabdfigfghfjbgfdeigbhciffeeafjehgj
jfgihffejcebidfgiefedcagebagieagdchffcfg

ebibhejajjgjcihehbiei                   
#gcgfbebjahcehijgafbbeacgiighacccibjghga
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ggbhfjhajcjiijhheegjd                   
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baibbfibgjeegdfhbchce                   
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jfbechhceeaaihdgbcdhd                   
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baiigfecggbhccibcfchc                   
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bbgdgebehdgdhcfjgbhef                   
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hibhabffafggeijadcjef                   
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edecbfbachhaabaiajhgc                   
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hajfgaadefhgjgiacgbbghjhfjjgddbiichheehj
jbddjdbeaijgiaaaedaia                   

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gigeaefbiedffbjigifdc                   
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gigjgcfhdfhegghihghfc                   
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affcfcdcahhaidjgfifdc                   
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fafbbcdahiebjhcdbbgga                   
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hiabafigcbjidfhceifjg                   
#ggadjahddcfejeifgchjfgeccdibdeabciijacc

bidgdfgefeihighbdfffjbgaaajeihhjgacdhebd
caadigdecfgigichibedb                   
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bggidfdggfacgbhficdad                   
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feihajcfcajdbacccdcgj                   
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ahgbdaicjajcceaifiicb                   
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cejeeedbjbcgfchgfcefb                   
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cacbadiegfaabfjghibbd                   
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efadjheejaiiehijccidf                   
#eaahjbiggibehidahdbeaihhechcicfhhjhjebb
bfeigdfabhcadcjfdjbhfdbhbicfhdeiahcaggeb

iifegefgjjfbegabgifch                   
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ccfdacfbjagaheiacdgic                   
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jggjdibahcjacbaffhjbd                   
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ffjdhchefjehcfbfdcgbg                   
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ijdfhdbfhacgafchbdfce                   
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hdifahidcjahagdjghafi                   
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jaggjjdedjjhbigfdhbhb                   
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cheehdbeajfjbejbcahbbdhfbegjchbdcgbibbaa
bjbgechahdgdgbcgdgbgf                   
#geddbgifjeabahejeejcefjcfceaebjejbihdbf
bideedafhfdebgchcdejheagabdcgjedhdbhdbii
fieadffhggdabefbfjjci                   

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ejiebgbhjiedbaejbfajd                   
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cbcjhdfchiiiagjdhfgcg                   
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ddhffcjeejjchbcjabchf                   
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gibabhfgbcjdacajcjcba                   
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hjcgdhadejbjejfabbgie                   
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ebhejfgcbahdjaghgahdd                   
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daiiecjgeejieiiiffffh                   
#bgefgcfgbjdedgafjjjigaadhjifedjjgihedib

gidhdjjihigbefciccbgifahcchjachjbgegfbga
iabbffbdfcgbcdfdiaiag                   
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bfgjbfjcbddefcjcgjehaajaiijefaeefjjhgbfc
jadbgidfadcjeejhjgdah                   
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cfdhdbegjdhdciidfbdfi                   
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hggdceaehabeffddiafgg                   
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idfidbgeffgigiadhceefbfdhbdgdijghcggcece
chhdaaefaebdcbifcfgje                   
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ccdacedacciedejgcbfgg                   
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efjheadhgggcdfgeeafai                   
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ecfajdhifjgibecieigaafecjaiiifhiiifefdeb
jieagfadhgbjaaighdfie                   
#gifdacfbicdegbfhghdigadacbdiiffahhdiecj
gbhcjhgadceihefbaggjihgajfbbcchajdjbibcd

jgagijdadbgbfegchbfgc                   
#efgeaeebaahaagifjcccjgjjahbjgdcjjgebide
iciceejfadeicejcahgeaghdjijgdbcieggddhid
bicceegaajfdjjjcbfagf                   
#iieieecccjfeijihbgaehidaecbejbigghfdfde
dcdfcabchbbgdgeeigidcagcjcadhhcbabfified
gjaiidjdgdhfdgaccaicj                   
----------------------------------------
-------------------------------------   
,the cube root of #c to #bjjj places4   
#a4ddbbdiegjcjgdjhchbcbafchcajghjajiehhc
iahfibecdiicejeggedfdafaidedafhgeifhbii 
#igccihedgeedgigjefdebeffhfhcejhjheddhie
diiffdbedbcidfaajbeigadhfhiejaegahebcgb 
#bgjijccbjbchdgeihddejfajheedjjbgbfjjhha
dedihhgbgeacfgceecebdfghffjgdgaefhhdcib 
#bccahiahbjagjchiihbchbbccbabifaffceejhe
bfbfgcdiaccejaffededhieghhageheebgdagee 
#iccfcacahgdadfgbjjfjdfchdfffdgeficgdcfd
aigeeegdidbdijfhbjhajidbfgabceijfbfegfc 
#fhifdfcgcfafaghbafeehdbehgdhbchefeiebce
hgaijcaifajdjgaciecjfjbhajbhecejhddcfch 
#jceaideejacchjiaebbbcijghdihigejiaicidh
jcfecaaifgdcdegjfbcchaaidaaahceefegfibd 
#hcbjjabcajgjaeiaeccbicjjdbhbgjfffcidddc

hbjdhjjaijabbdahahjegheaahjbghfcediibja 
#dhicebcebgifhahjajijjfbcfhcecbgigjcgcgb
dfadefeagcdaececcijiijdfgajecjdaeficgfi 
#jcjeadidiehiiebafaffeiaaffccchjaiedbbgb
ffdhbhadcaahahddagafeecegffhhahcbadjehi 
#ejcbgbgiiabgibhjbfihcegbacefgfcjdffgfca
djihbficjifhfbbdgfdidadjdfddhdbhhgaccjh 
#giidfhdahgjjjbjdefahgfijbgejccjdfbjcffe
fdddjgagijiaaifihjcigdgdghhhchjbfgjgbbh 
#ddgdhaeidhbjhgbcifaafjabbgajfgagajfffab
ghahacbjaajhaciecjjigbbgbbfffaiajgjeici 
#ijiijaaiabjfgeiiiiigbhcibcdjajffeaabjag
bfaijhfghchbfdeedicjfceacaaafeggbdfgaah 
#jacffaihgcijfajhegbddbbdihhiddefjicebhc
hahjegegbcaeigiaidhggdccedaegccefbbfcaa 
#eiedfajchbjgifgcdfidaejcacejbedgfjgceea
decdccdbddfaafecadcbbfaejaddhecdihfffhc 
#fjbhcbihjefcfbfibejejcdcagfehcfghjbccid
iggdjeaegjbdcbhgcbiefehgfecjbhfgcedbbfb 
#caiacfgbhdjjaejjfadhhedggfbigbijhehicgb
gehjgffefegdabfiiiafiadjcdbafibbhgbjffg 
#ihiafdfjaehahabhebibbahefcbbfehebjfadhb
gjiffificgfadffdidbacacacdgffcfidhabjjg 
#eedfdacicadiaaedgbgjhagjhjdaffihfaedcgd
adjdejecefhfdifdcfidbhfedfhbfhjejcbcfbe 

#egfigfcdjbjbdjjahiggdejihdgbhbjicjgfchg
efagfhcdadhedgagjhibbghgighjhhcgeaiifcj 
#ibbbcjgejibjgiiehedbdcccjeefjjecjadacih
jjbieibfefajbhfgdebjijcifgjgjgjeaiccbbj 
#fgjjjjfejcceafdggeecdcbfgibcibddcebijaf
dfhdjgaibajijdjcgeggjcfeicdbifabgidgabf 
#cbebjacbfheafgebdeebicegifiiegjdefeciji
jahfacfjbdejejachefggjibcbhdaceedahcdja 
#cjgfaehiadacffbgcabfigheddbddddddhicbga
igfcbadbdcjafgicidchijdiaffgdajchdhdeec 
#agcjibjihhibajjfddeceeiajfaiiifidhciddd
jibebieiijhb                            
----------------------------------------
-------------------------------------   
#b99997a_/c7 to #bjjj places            
#a4beiibajdihidhgcafdgfgbajfjgbghbbhceje
gjbeadfdgjaejgihjjhaigeaabaeebiifgfeacie
idhcgbicifefbdcf                        
#beejidaedcajbefjcefaeffebeiciijbdjdjfac
gcgbbhdeiaajcjdbficeebdfifjfdbfaffbejjji
ggdgdebfefedhjcjfhfga                   
#hedjeeahfhibdehgbeafgfdaiicgcgjifiejihc
hbghcafaciiaeeabicacfiecffahcidgdfcddheg
fegjcjcaaijiehieihdgd                   
#aajeihaafbijgjeceijhafdghjaadgcebacbedh

dggabighhjbdbbjhehbjecbegigbebfffbbjbffi
jjeffefjhaiidgaefbhag                   
#fdjejfjffdhbfggcegbfgjdaidhfbjgfbaddbif
efidbjejgicaiagbddadhjibjddhbcbhdjabgdgj
cbaifdbhbjhabjaijegad                   
#ahhiifdeiiihcagejchjahhhfhieidbjbjeeibb
jbaaedgbiigchdhhjbfjgcfcfigdaghhggibaegi
hdfgejiiecifcjjghbfji                   
#eifbdbjcdhcbchffjacihegcfcdccijigcgabfe
bgiieiiaifiifhcggiacafhafhaeddbhhejbgife
aebibghajgfgigadjjbjd                   
#jfjefgdhjcichefabeagahcegjjfijgihdiifcd
aigfbiadgdjddhcdedjbfigaedgfbbheacaghjbj
fdchghjdgfdicbbegijeb                   
#hihdfgjhehjebhfbehacjjjedbichhefjgbjfji
gdgbbcjdjfcacegbcdicfdehdjfegeiafiafiafg
bgjfjabddjbhifgjjjjaj                   
#fijhajcecachebijbgjjdaejhdbcbccfbcihhic
hfdifghbaidadihchjbgjgbiegagfhabhgijjadd
egdfbbgadggjbcdhcegae #aijeej           
----------------------------------------
-------------------------------------   
,zeta7#a1b7 ot the derivative of ,zeta  
function at #b4                         
-#j4icgedhbedcaehdcgecgjbegdjidefghfdigg

highfjbhhfadhbiibehhedccdhjc            
#fjddchaacabgjgebbgicfhidaeadaaeaeagdica
achbaafbdafchecejeidjdaga               
#eifagccbdgaigaheagdiabdjbfhhbadddcgjjaf
cicajaejdeajgafjcgcegdhgc               
#acebiefjegacceebeiccahjejeadhgbecdgiiih
dgagcigegjcageejcjbfaijgc               
----------------------------------------
-------------------------------------   
,this number is 7exp7#b7-g7_/b  ,ref3   
,francois le ,lionnais1 ,les nombres    
remarquables1 ,paris1 ,hermann #aihc1   
pp4 #bc4                                
#j4aidebhjdidfecbeaacfaebacgcjbhgejcijfe
ijaegghebgeibcffbjdcefciaab             
#fabhfhihjciebhhhafibaefbdbeciefjhigchfh
gfehjfcbgcidccjfaidbcjahd               
#fcijfcffhgbciaifajffiijchhhgdejjfaddghj
cgjecgfheaaieffedgcggecda               
#jdcbijiajacdhbcicdajdbjbaajdiaabgfagdcg
hgabcabgjgjgcciifdjfdffei               
#ddjcechafejejiffhidihgjcfdiicdhjjdgfeaf
ecgegffjdjidaahdbcdgcifea               
#diefggichegbbhdaefaifafcfchbcjabidafiii
hbcjfjfdbdfdbfjdhcidebefi               

#dabccaiicefadjfhfcdcjecbcfghacahifdgeia
aacibadgdbagbicjfgfdchdfi               
#ccdibhgfjjjggdihjjgdjcgeceihefdffhdgjid
beiihfadddacahabgiheigjed               
#giccjiegciicegebafdaihhdffcbcjeaageehae
faidiiejjebefhiagjcchbbdi               
#ccaidfcdbhjgiajicbajgghhfbdbdfjjeeifgfe
hajeheigehfehgcfcdhihdadf               
#gbeiiibebgjibdcaeihefghdbigceaadefbghfi
iaghjeebegdhifhdjgbeachhb               
#bdjchbadibeebjiajehebgigbjiehichdagcefd
bfchbdhijdhefgcabebjgjaag               
#fbajgicgjdficcdabgacegheaifcbbfdhbjbbge
cadchijjjfeeedfcbejfibdaf               
#gbfeacbcahaegjghjbceiddjehhbhigacicagdb
iiechcebbfceeifhfdgicfaii               
#giicecffeedjgdhjfbfeedhhebifgfejdiebeaf
dhdjecggajededchhacaedbhf               
#bdbejaiacicfachjgbdcccgbebhbdicfibicheg
hgeebhabagaidgaihcefigbad  ,the         
,dubois-,raymond constant to #ajbd      
digits  here is an expression for it4   
#a_/b997;exp7#b7-g72                    
----------------------------------------
-------------------------------------   

exp7#a_/;e7 to #bjjj places4            
#a4dddffghfajjigffaccfehcciajheifdcjbbcj
eheiedecbdbbecafehbjebbffdcjchedicggahf 
#adejeegcehbibcjdgjihheaadbiebcahddheege
daihjcbbgjejfddejgdcaijchbdeaedcdjdecbc 
#cdhbiiehicedfjjefejehdejaeebbibafheecef
aaegfchhhigiaiabfiebgjgajfhfijdeeddaibe 
#djbcbgfcbdebhbiaaeaeeecabiddhjebfcieaia
ggiddjhafgecbjjjaibddgcjdhiijihfgbgedje 
#ajieafccdfedcbahfjjcaiefgjbihbijidcjaeh
hjabfgcchjccagebbhbjgiabhcgdeaajbgjdhgc 
#bfjhadighihhihcaijdfcceaadcheddjeddgbfb
hafbgdgiigdidfjdhabgjceffigijhcffgjgdcf 
#bhfheggdedfibhebdbbcieegffjdiabaifgfdgh
iffejdafbdfaiigjchjbchcicbgjigcaheiccgf 
#gdicecghfaddccahahbbbhcihjhceeidjeihjhj
adihcggaeihggcjiadbbacgffegbbhhicjhcjch 
#feccgcgifjhefceibgdahebdbjgejagjjieeejc
dgccghfgbfccfejagbecbjgaifdfccjidjcihje 
#giaahabaacbahbbgaegibgihbdihficacaeabeg
fdedcabeffgbfijghedcjigcjacaigiedjjgcdi 
#geiiaedaabhdaieaaggiejiefijhjbjhjbgfgjh
bjfbbdiagjbhiajejdiibeejhehhchcjaecjiij 
#afhbdcjiddabfbhbdfgfibcdejdjcfcjafbebij
gdbgbbbfhfgiicihbageigjhbjiieiafcdjiffh 

#bgbffafgebiacicihhgdcighjhfbgddbdbafibi
jcjjfgfgdfafjihfjibadfecahgegcahgcjejhf 
#abfbcbifebedccijdijgdfgfffdhaiajehjgcdh
ddfgeiddfjhgabgbddigijagbdbdhcdfjdjdagi 
#cdagdggadbejjejihiicafafeigdgjjhaefagic
cdaifchcijfbjideibgegdjidehaaaggdaejabc 
#djajijgchhffehdfcggjcdjahjdfihfbiegibae
jhjibjiediiccdjceeeiggfdhehahcagddhbcec 
#dgcdeehagbjhedjfgbfgajdffgjjgdeiiahifgi
gfiiecjcbceaeecgjcdbjagaagedecigdaffbdg 
#hjihbgaihadcbjcibfahchbiejdafhficahdbdj
jhacdjjadgigcfhcccfchbdbhbfaddeidggcecb 
#hiecaefdfebidbgiicbebadjidddcejjajehaee
fjbdgjbbbgjgjcdcaedccdfghbdffgcijcbjfgb 
#bihffhcdjgeeegegbjcdajbiejdfgbiiieehhia
dcbgfedjbcbdjiiijhcfjjiiijfjfhbediidifb 
#bjefaddejcdfdjfhgcieddchfijijebjhcecbeb
gagecjbdjfgcebhgjbjhddjjibdhbggcfiifdbe 
#ffedehechhfihibfggabgiagcjhejgigadgbjig
ijahaebcfedjdihchefcdhaecaibcgafcfcihbc 
#dhiabgfegbadagbfafbghfbddcbgjciehaijabc
jabeiadeicfdfcfdeggjjfhdcjcfjiieiafbadh 
#dgagfdbicjijbfedfdcebagbcbaciijejgicbci
edbdadeaiiebfcbgaehcejhcecbbdbchhibhadh 
#jfbgjdjhjbcfdfhbjhdahhagifdjadjjgeciihd

jabadjidjfcbciheebjjbjeiicahdegibegaiai 
#hgjegigfahdbfiihjidegfjhdjcacgbhbeghbib
ejfjhagbecec                            
----------------------------------------
-------------------------------------   
-exp7#a799;,ei7-#a7 to #ajbd digits4    
also called ,gompertz 7#ahbe7 constant  
int7exp7-x7_/7#ax71x#j44;infinity72     
#j4eifcdgcfbcbcaidjgdcdajghdiicfibgicgfj
gdagghfjaebedhghaegcdhdiajd             
#hbcbgbaiaadhgddagdgjdcjdigjicfabgfjcddb
cgjcdgdhdbhfbcfhihabjghbi               
#iebijegaiffagcfibbbffehidjbdcaheaceadcf
hbicgfcbifbedggaahgigdjbe               
#bdcbcjbjebaaghhegcghefaggbhcfebcfeacghe
eidhfgdbecefbahacjjhabjhc               
#cghdbchddheieihjfffihceicbaghbfdhifhfjd
gbcajiiifdeajheeehadaechc               
#ebjfafbegejjhcahhgdahgjagehaeahegicajje
jfaafjdceebidefgajcdjaejc               
#fffcfcejbigeehjgadaifdfeibjecgjfjbefche
hgedcibbcigfchcicbgjifahf               
#ceeeiedbjhadaaagbdeicchfedfebdieebggajh
ghbiiijiehjcejibiiagiafba               
#fchifcefiaceejfigcabeedhiigieficgaicjga

ghdchgjadfifgbhjggeaghagj               
#jdiiajffjeddhdgbbedidfbddacgjgbefacgibh
dijaigediihcjjcgdiebihcjc               
#hdbfedgfhbdecaaachiffeajdfjfafjefihgjfc
ejfhcdgafahicabddiabcjebf               
#dadiiahahdcdchbggdefdhhjdbhaidfbfefiadc
hbjhjahfggddddfjagdhcacfi               
#hieiaebfgefdghccfiedhgahfgdjjiibeifjbba
cajgghfaecghahehijbafcbbf               
#bieffdbjgheabihgcbeafccdhdhgehhcdjbefhd
dchigejgdgidchfaecadgibii               
#cicbhjgghdciihhagfiehibaihbfceggdjfbcgg
bafhbbhjegafiiafjfifccjjf               
#fhcghjagchbghccifcbedddbfbjigiidadbbicc
gchefbhdijgiffdbijjehddjd               
----------------------------------------
-------------------------------------   
exp7#b7 to #ejjj digits4                
#g4chijefjihicjfejbbgbcjdbgdfjegejjghaca
hjcaeegjeeahdgcbdjhgabghbbebbegcgifjgij 
#eggfcchdcabdhejgiabagidggcgecafabfedghh
ffabchhdfjcfibghabgccgddghcibbaccihjggg 
#gdijjabbhiefjgdajgecgjbciaccjidgeejfhbj
hfehahbjbfifdghfhbjhdjdbbjihbbeebcdhgeg 
#dbdfbedadadfgiibhabiccahhhjgjgfccjajaic

cghiigdjgbiihfifjjiecjccjgeaecbjhahhbcf 
#hdfidgicjbiiaceehggaddefhcabcibcbgbgfdf
jbehhcciiifdfababhdibhebjifghijeachhbdf 
#fcihgabbhacgbfhfajfdgcefbfcgibieahbbbgh
dbidhdcdehfacebhgfichffihegebjjaediifja 
#dhjgejgaigabiccfidahheaiigbbhhhbfcfbeei
gaidajiehffaiadgihgaejdcbhcigficbfdfajb 
#ceaafcabchiiijjajeacghcdjfgfddihffchibf
hefaehbahfdbaeeggbdhdibjaaaicecafbaagai 
#eagcagdgbfigifhbicdeaiihejedahdhfcaigac
efheidgjbbiabeegcihcefaajeadigicfhadejb 
#ggfddhjgfdbiheajdahbaagjeeiddaiaghgfhcd
gabhebgfdighjigacdfbejdadjbcebdbaehgdji 
#chffhbedbgaegjcibfdebifdjdeejfbhgghjjac
aajibfejachdhccdecjbfdfcfcadaefjdgahhha 
#agfegidbghfcdheiijgfgjdebgaaicgbiehgbci
ieihgjgccajhadifabecajiggjeicecjjiijejc 
#bifhajgedbajijhggfbfcjhegbdhejjchbghgbb
gfaddhffgdejefdihgchehggaegeajefbdcdchi 
#dcifgaciddbiejibfjjffghbifahaifebhfjfci
feigafjicciehcccecabhbgcgebgfgfaegafhjg 
#cbaieafijaifdbjgbdeghhddggeejfiffadedcg
cgiffgegchgafhbhbcgiheegegaibadaiijcdbh 
#bhfgbbhidiaghjifefdgbgbcbdecfjdeeahcbgc
fhhcdaigicdiehaadediggjihgieidigjfdgehg 

#ggdcefdjdghhdbjghchdjcagbjjfajgidicgdih
eadcdbhdhejgedbhahabahbggfchdgajgjjhibh 
#dcjjaccdccdgdjfjdabjgijajifbjejgifjjiff
iegbeffjffagjfebjdhiifhbbgegddccbbbifhi 
#hfhgghadacjefgidajgadbdhgghbcjbghjcefig
jfedcficjfehajbbabjeicfegddadhaaagagcgb 
#cbbjddcgfbjghheggefiaggjfececjiggdcebia
cheegibjhfcficacfgbeficdiidaaiecgdccffh 
#jcahjifehgibedighcbdihaidcaieidhddeiaai
jhjgbcfbidfhgahhabcadcdbdgajgiegcahhjei 
#daafbiafhffehgghiiahgeidbhdjdfhgdggffjb
iafgifdbfbgeagfagcagfjbifghdcabfdhgebbh 
#ddiecbdajejchiaafjdhfcbfcjiehjgabbjgjfb
hjjhjbfacihhafcgcjabcifahabdcfheifigjgc 
#fhghigeabfiedbfhiaiaggahheadgbahbccjejj
bfjgbacihfhjjggaacjfeajdjdbgddfabdcbdah 
#fcfechjjehbagbddcggafccadggdfajcbafhaib
iibbchhhjdbbfigjgjjedhdfgdaegcgihiagcii 
#ffdfbiffgbdcghcgbeeddfidihddeacjchbbejh
jjgfeihfjcfedahgeebiffchadjgjfdjeighadi 
#aaciffajgcajhbebjbagdgccfjdiacgicagbfgf
daddafhgbjgjajjicichajjgfeicfcgiacfjfef 
#cghhcjgdcdcfjgccciedijbdabegcgiiajiaebb
eahehgcddfhabhfebihccbhgbbaiajaafibahcf 
#gjcheabbdehfdgcibdejidgjjeahjfajefgfiab

cihcecjagabifcaecacagibhacjfgjhgcfgeeab 
#bicihifgbhefiiceehaieeieidbegfaidjfiijc
cjagejiafhagjeaifjfhfidgiebeifgghghbaaf 
#icjcjheifaahjchifgifabjchiahjjbbejijcge
dgabfibgeafjaijecgjibaffjdgibbdjdgjhcji 
#dafbdjeebhjehbabcdjcghbedjejcafcehhdgfg
checjeabiedaijjadgifhjeidbfbihbdbgggjag 
#gcdiifgdeghfjjbjchegegdffggafbjgjijaijb
bjagefegcffhiecjabjjjfajcgjcidgdebbffij 
#cccdijbbbcbifgdhdgjahchhjbbfgbfdacehjhf
cbccififeehddbafiehjifdhfeihhehhcjghghe 
#ifdghgidebgdbggcbfiiajjhdgdfaaggcjhgbhe
bijhagaabdjjecbhbhcecebbghgbhfcdibihjaa 
#dffgedehfdjggbcaiaeijjjfeefdagcfgbiieha
afaffjheegdihgcdccjgeeagjcjifiafgbhjfie 
#fecbjbffcbjijdgehccgdedgcfjdcfbfhabcejh
jaheddjagejdjdcgefjhjijfcfaggaaihfbedgj 
#adijdhajdcgibgegidibaaadghcechhgejbiaac
hiicgfdgejjefageadgchcgfajfhfhcdafahddj 
#afdiabdiegihifdchchjhhabibaifbefadiajea
ijaabbiaahbcfabjfeeadcaaeihbdejbaaajhge 
#gabdgaeageicbahjdiiidgjafahdejagfhdfhii
jbbbfahabfceggdjadjaihecighicbfdjcbcghf 
#chebecdicfdaeeafddjigaaejhhdahbehbfcjac
efchgifaddbcjagegahebcihdiihibadfjfgahb 

#fcheecfhigbifhhebjfdiaefciefehiaiiedaad
ghbjgeeaahcahegdbfhfjabbcgcdihfgdhgjbfi 
#cggghdjdbdefbjcchcddhgaaidcciiabbeedbda
efgchhafeffdhccafhfhediaigajbgjggfiffbb 
#iihaccjhjabaadeefcjabiccfedbaacieeagcga
bdighjdefcjcajeijafaifebhiigeaaifbcdhdb 
#jccjgdibdifcahgfddjdeeiaijfecijaddbceae
fbeaffacibbdgggfgdeeeggijjadfabihjhcbdj 
#fgjefaegjdgciaehjjgjfbdefahajbahhfehehe
igdjdjdefdeffchfbfbaidebjfgccijcdhheicg 
#ighadfieafbejhfficeaafafgbdjicdbiejdifi
idgjhefcgcchiehjjadgfhecjfbedfajigcjgdh 
#cgfcjbbabjffhdijcjbdejjbbdejdaaehcebfbc
eififdhdbiebbeiieciiggdhgedggcjfjcgeaca 
#dcbcggdddaeeffijidbecfcbbfdbebhffhaefcc
fdfegjagfhedecijdcbbeafjggbcbhcgdadbbce 
#dhidehhjfhebjbhccggijjbhcgfcebjibgdfdai
gfafggjiicdfhhgijdicgidjbbaegafgbdciebb 
#gdjacbfiahaedicdcdcbhacffbgjbdccfaiegbh
hacfehachidgfhigeacfdceefihgdjaihgcjbjd 
#eajecbaijbajaejjaagadeijieadbijaacaifab
chebabfdbdhjcbeagieiidgcabffagcgefbhidb 
#hdeacchgjgjbbgjieaieijbbchgbceajiacdibg
eaededecjadbeejhicadajjbiidgfgfjhejiigd 
#jiidiefdcjibchjhfhdhagdfddehicaejfeifbb

dheajbhihhfgccgefhecfefbjecaacbhebghbhj 
#ceadcgacechaaicihecbdhecfffigjfcjhigcfi
jjhibfgghjgdghdcehdcbhjhafheibbcafajjbc 
#icgicehfbefgebdeheahbbacaggdabdhfjefeaf
eecbagjcacejjbdaeigfdhgjaabiaajjcgdecgi 
#ddafcgifgffgegahbdcaaagbafgjcjbgceheadc
ejjccbcdcahaiebhfdjddifbbfejdgdeacagicf 
#cgbjhicidcejhfabchjaihjdhidjacabdijcfie
gejgeigiabbafdheacibbghajafgjbdddjjjeha 
#aiieehbfebiecgjgdcchcajgihfhffjiebffbcf
hjabhgigibddfhbjaghacdjjdidbgjjcdhbgidg 
#aicgccahajdbbdafbjhbbdeegidbeebfehfecjd
cechfccgfjgahjgcfbadicdddddbebcgbjaafbb 
#aeghdfhdceeeecdfjaacacfbdhfeedjcedafbhi
acfdgjghbjjjcjebabicaejhefdaafbigfaebfe 
#igighefiicebeiidegafcbfcjgaegcbaichaaff
jffchiieaadhjbbdfffffcfgjgifcgbecaigijj 
#fedbbceiggifaabdcjbbcheecfibffeadddcegd
jbgaiefhgeddfhhiefiehaajjjgihffjbdcbfag 
#higbafbhebbgjeaejcjbjacjcdaiafeghefhdjd
fjggagdicjjieigdefeecfijhbaghdhedddehga 
#hbfghfchf                              
----------------------------------------
-------------------------------------   
exp7,e7 to #bjjj places4                

#ae4aedbfbbdadgibfdahigfjdcjbgbfbiiaaije
ebhedhecfhefacigfiadjgdfdjeiadhcjigcgcj 
#icddcbfjhdefifhceghgcdfjeaaehgbfhhebheb
biehdajhcdibffdbffegfdiaahggidgigjdaedh 
#ajdfagfafbbichhcfhdedhbaidcbfeahhbcfihj
fgehaacabcbbiijcedfaccchcceaheifeiedbaf 
#ebejgbjdhgaacafidhdabdhhcgjbhbihajafcji
djdiegdggiaiiacgbdecbagbhechgcbaiajfhji 
#ggiadgccfehahgfiiifgfidagdgghfdijchafcc
ijejefabjdiggfabecdhjeddfffbifjgidjbjai 
#ebihggbgeaheecjhgifgggbhahjebgeceicaabc
igeijfjjeahhhjhhjdaeagfdaedbfcbbgfecifi 
#cfidaibhafhadahjdhhaajejafbbhegacabeabe
gcfhfjhdagjejbdgecgbeeafbedgbhdgeadajde 
#giifdiccdfdibehcgggcbiiggiiebfgdfbjgjhh
efffbeggidjdehieddijjieafdfahhejcbdeaee 
#edcbgfajbeeacgicccgahjhedfhdadgiaggacbc
edgjejfibbabfadfcfjacheahajdhebiejffcce 
#ibjegeedadjjjicgbhhacbgeffaaggigfjdahfi
gcjafifgbdhgafecdbibjiicfgjajbaejdjhhbi 
#diieigejffagfafgaihdbfaeegaaaacbfbidggd
abchajcaeaeafacjffcfjbdbcjajgiifgjihjhb 
#efbfibjdajiddfegcjcghihdhccihhgbfjaahig
heddbhdjhdecegjajagejgihiahifeadjcibdbi 
#hdedddhhhfjjchdjfjgajegdcfacejibfigbedc

eifafjibifgdhhfabjigiajfgjhidchdjjbhece 
#aceddjhgaciahhahebcfghieehehcbbhdahfigf
eejbgcdcffdbiejeicjafheijdafcdifdhefjic 
#ccfdbjdhciedhbjfejchdibdehjaicidiiejggi
hbaaejifdhgbcfhcgcjdjjfchgjjhcafhjiecfc 
#jbghbehiecgcbgjbcajcjghcghdgjfeeicbcfga
ajegjbbfeahhgbjgdaabhcdihhhahefghjegcij 
#afaiahejbjafdafjbifeadecaifbbhjfiigegfc
beedadhhgbbcadaaigcjhbfjiicfhjiieeecfdj 
#hgfihgeeiaebgdagcgegaaaehgefedgehfefhjb
agcghdihafciajahhjjjcbjbbbidddiadiiigia 
#abahccfadhadcgjbbdffagbibiigciaiecbjhec
aibdcabbabjffbagjbaedgdaebijfgeacgccgdc 
#ajebhaifgajhjggiidbeijaihgihhieeffhdfii
ieiejfdgidahcdjhaiadbfeedhbhcbbcdifhjjc 
#jhaigbfhiegedffjehbjfcbjhccjcceecagaiij
dcdifecebbadajhggdagccbecifccdehjdjjghf 
#abhibgahjcfadhfbaeecjfahchifggadggdfhad
eeagcijcihhhabbbifbfdjgicdbaeebehdggdbc 
#jejhidedhecadcjgecbgcgjfbdfhhhadgdbjhga
aabfiegifcahhjehjhidjjidedehdigejhjjjch 
#dedfjefhjjfcggbecdhajbiaaceaihhcbejeeid
hcjdgcejhcajfjaeeediadagcifgdhbjaffifbf 
#dffdgbfibhafecbbggjcihedcdagchchgbegbcc
heecgbcecdfjifecbfiiecaiieceddcfjdaegef 

#fdadjacebaaadiiahajaeihefbhffdaabadbeja
ebdjeebifdgi                            
----------------------------------------
-------------------------------------   
exp7-#a79999;exp7-#a7 to #bjjj digits4  
4#fibbjjfbgeeecdfcechfedbaiigahbghigfadi
jfghjbibigeddgceichiadhiiifeagaeeijbijh 
#ecfbabcjabchgfdicecjihcdgfjdjjdibaeacfa
edaedfhdcijhbffegdbdccgaebegadfefeeaghd 
#aaajabaahjjdehacbefhdjajibehicggadhcfah
ifjdhfjheecgcdchgiidffebgifhiihecghgjdj 
#djjbgeceejifehhhhgbccggeadcejadfjjafghj
fijjcfhjacjhedafcjaaaeacbcghdgjeadbadca 
#gcejjcgegegejbghbgfeghijiajijahjjdaihfh
hfadgfhehjfhabggbhedfgbdajfebbhffafgagh 
#ecccidbjcihffiieabfhddjgfchbifgfahhgfia
addjifgfhigjifdggjiebgddhadacgejhgfjbfb 
#cdgajhbjbjifcbhcfdcdehfifbjccdfbjiichfb
eaddhdcjhcfgefjhhajjcigbbchcjcdejfedjaa 
#iiicjbffjffiehcgabegecijbbhcceifccdcbbc
cageihbijbihcbebifgdaehjjcddgidbfhjifag 
#iafefbgaffjbgjcgeahgfbiciibcdhdjfdefafh
jgbbgbaghfbfjijhbaaccgeiigfigbdedicifgf 
#bccgjedgagfdhhfieiijiffacdcbjbehcdcejjb
ibdjeabhbigchbicfjhhacaigcceejbdgaebcaj 

#faghghafiicaeajheifgjiggcjhcaifibdigdei
decgghhdcajdggfecafbgaciaagieahbiiebggg 
#ihdghdgdhjhcajjejieadfhiahjejgfddegecif
hicjfgdbffjigdhcbgcajbajcejcdhhfejigicd 
#hdgajhjgiehhbcacgecfiadcjidijcihceaiiag
bhgbaaeegjbhefacjijdddahjahgdhgbghfagbj 
#jdibdffeecdfjbdhffaicjjajidghbfdceeajfh
fjgfhgdacggidghcgejbfjhjficgibcgagdeffg 
#cbfibjicagchjedjeaeedbdbbdihighchbigadd
dhihcggcdgacacgcbbbhdcejecefahgebigfbhf 
#jichieihifcijcbgbbacjdhjfghdiigaeagfibf
ejcdhbcjeacjdecibbjiehejhahajaiecbceihh 
#haeeiegcgahdbhafbfdhdfdifheccibhhigcefe
ddehbbcejbgidibjieddeidjfhjeffjhajjcbei 
#bfhagiafiaacbgfichahfcbcahhhgjbefdijbjh
hecacfihcfjfgabddcadchjcgfbibbdjcfdahca 
#gjgdadgaijgieicadccadajbfdfeibedegdhbcg
jcjajaegjiebejjajjagdiihfegdiefgdecjacj 
#igfgcfaeedbdbjdchcdbhgaadifcjdafbbbbcgi
fejgieggecfhefbbajfaaafbfdebefabcbebgdi 
#cjijbfgchbcfddcfegbedaibhhfcahdgabdidie
ffbaacjafbbfifeibdjcbehgffjjdaeifbbfheh 
#hgahdbajjjjebbgdiifcdefcaaijadchhfdfgfd
gidfdhbhehajjaabideehagbdabehbbfihiffcf 
#adhgiefhgeageacdbbhjcjicfhcifjibaedcghh

gihgjjggijchgdjjhjehcghfgbbiaiidcgjbecf 
#hffbhdhadifafdiedjbdijgdhgeceaebciiicje
ibabaaidajfjjcgiicgccegajgicfhfddjajhif 
#eehddaefhjdcbiaeiedcbdjdiedcedajhddgiij
aiechcgagichbedccddjhiabfbgahbafceefjih 
#fibghgeahegaiejffaceadjggjjfjfghhgejfjg
caadcgjbheeg                            
----------------------------------------
-------------------------------------   
,the exp7gamma7 to #ajbd places4        
#a4ghajgbdagiijaigihebcfejdajcajgagiedia
fifdebadcjcdcjbjecegffehgf              
#eabhdajgfhacehhbicgjgegdbafdhhdahbhjccd
hbbbdebbeadegdbjjajeegide               
#gdbdhaifejjhhaefhegeabfdejjaaehdeiegbfg
djcehbhaifgidbijiejfiaegh               
#ddebdddajdiejfbdgdidfdfgcieddbbdicibjfa
bigcffgahiibifaahaghagafe               
#bhfddbjdiaifchhadhdabbafheigibaajgiccdf
dbdiaifbdgceehhbbfigiaiji               
#fgjbiaejaedccedhfjhficcficcgjedfideeija
fbcbgdcebiecbeihcgbegffje               
#gjcfaheiiaebbdciagghjjbdfhffjffcehfaagb
hgibghcgaibcaacfggegcicid               
#ajdjiigeafdjbjcfdgcdhdcebchfchbjbabbfej

ffdbdgfifbejjbadgbfcdddia               
#ddhdcdhhefdbdaghigdifdfgbbgbhfacdgchhaf
biijhacciihfcgfjafejieabb               
#eicjdgjcdegddgeeiejfahhiaddhefiibciffjg
cigeafbaefcdbfbhfedieeacg               
#cijibehadaifbcajgheeaabjbjhjaiahhigdijc
bgfbddidfeciieiagcbjjbcgj               
#bcdfigbeiehfhbfgejdhfdjbdjeagcjhagcccfh
gdjejjidihajeigfdehfhgjbi               
#chifcfgjfebdhgaefiggjiijfedgdbdbaahjabj
dadejfajffeahhaehijcfjehb               
#cidagiheaifcdbcfcgjacdggagfcfdfahhfhfhe
ffhiahagihhbaijhfjadcjjja               
#ijffgbeegdfjegabgbbafiiaadahiccfdhdfagb
bdheihhhjfigeebfcafefjfdf               
#dhhedjbhbbghfdjjagjgadhbiebdadcfggjfiea
gdfegebbceihgaffgdfbgcigi               
----------------------------------------
-------------------------------------   
exp7-exp7#a77 to #ajbd digits4          
#j4jfeihhjcehdecabecgjgfgijahgeifhdfdbdi
cheggjdhbebgifdcfdjbdgcedaef            
#fgcfccjjcjgefcjhajdjhhbdbdeccgadfggdefg
ebfecfadagcheiaebfhabiggg               
#gfhbiebjiidgdcaieejcgecaehafgbhdghfbgbi

djheddacdbeagcfbdjdjbhacg               
#cjgecaifgijhjhcfhicccigbafaajjgacbjheih
fhfgcgccbidejabeaaghdfeaa               
#aiigcgajcchdahbjabdijhghhhbdajcdjbejefb
ifhbbgiiiaiejebbjjgafhgfi               
#ibbcdjiidjhadjigghgfbfiecgjgicbagdfebbi
fidaddggbhdjhcacdgeggicfh               
#gbhfdhgifihcgigbabaidbaadcfjabfdjigafgg
gdieaahdfgebfhidhiihfdghh               
#hgcejcadjjfabajdbbabgdjbiiaccjdgichidfc
jfdjieiifbhdbeeaaagdjjbag               
#abbdcjaebcgfehjaedhcfddhfbffjhdiifjgcij
aifiahcjdddeacabhfgejaccj               
#fgaacchafgedjjdbcfhbihecacjjdgcgaibbdca
cfagieebfdaghdiahfdccjbgh               
#jgefaccddiaffjfebchfjijhjbbidfgifbhebcd
ahbfdbfeehbeefjaabbcfdjha               
#gdiifghjaedgcigacigfgihbfcgegccajbafbib
bciddbcggifbagachhhhcgafc               
#fedieggfideccihcdhicabaigaidfdcdihegfha
ggdehghffagffbcfdehdffief               
#ejcjjfdfhhdjjjebjdficedhhaehjihcgdbebga
djaiehbffjgjhahggggfadchh               
#ddajffigahhdiagiejffbecjiiejeeaabefjchg
cbhcjbfahgiddihehbaabifgi               

#ciggbjhhdjbhfiaghbibiiafhceeiacgegjjahf
edbbghjggeccbhfbjhihigjfa               
----------------------------------------
-------------------------------------   
exp7-gamma7 to #ejj digits4             
4#efadeidhceffhheafihbdadcbadgijhhjghfgf
egajchfibeaecafhaedaeijgfjdejhgifgjgdbh 
#efcgacbhgaaehicdbadcehgfgcaiacajjiedejd
ahchaebidifdgfeajdchebjeffghjiaeacacjeg 
#gdgiehbibhgjbfjjcadadhgjfdfeddhihecfdje
cbhdggbaaehjedcaeiehcddfcgfghhcgdhjajcb 
#fbfcdhbefiihhejjdjefeaghcfjccigbfbcgdgf
idheecfcgahheigcbhdcgadhdfhibgjbjiffgab 
#cdjbdjgcegaghebgdbfcbfifbahgihiaeaadgcd
ajeceifdciedcifahjibidehhcicjahaghgajgd 
#gbejecfehejghhhfdccbcbhccagfbdgghefadbf
bfbffeiajfiecdfggifidbddfchajigcjbeeaaa 
#dfahfjccadchhgggbfggjafffcidahchi      
----------------------------------------
-------------------------------------   
exp7-#a7                                
4#cfghgiddaagaddbcbaeieebcggjafadfjhfgdd
ehaaacajcagfghcdejghcfhjafigdfadieg     
#ddhiihjccegadgbgdcdeiaifdcgdffbgcbebgfh
dciiebjhbdfigegibgijabijjhfbffecehididj 

#ihghcjibaidcfgcggcchaaejdhfchiiaabeadef
afcddihggaiighfhddgeiegicigdgcjbedihibd 
#iedecbcicffbjgifdhajeadfdgebjfabbidbbcj
hiafdibfefffjjcfejgdeggbhcgjeecbhecgchc 
#hhajfhjdghgfaaiefhbihicdeddigcejgcicahe
iibaffagdccjjcefiicgbjhbjgajbbggeahjbae 
#hdiidbccghafijgaeffgfgagfbccffjhbcjcgfa
bbiaefbcgegbjidgjjjgjdjejigccdbefggegfb 
#ebebhjcjcgfhhfafeaegjicfecgiiedbgdjfcgj
gaghghddedaidfgdijicacjfihjefjafcgjbaaa 
#chiggdbbhbadjagchjbcbhcbdfebhgbiachijjd
ffjihffeieabdddjigfiiheadeiafdbhghjcgbj 
#bjbeajbbdeghgcbaaajeiecgggfhjgdcgaabbjf
bdjjjeafgifebhjigedddghjbhfdhfjjfhchefd 
#bjjdccfhdffbdhdcdichfiahbfbjfbeahiidhba
igjiibdbcdbebjgeajdibjicddebheabddhfjbb 
#deachjihfdagdbajfabaiecfcfhcajjghbjibbd
hjdfecjgihjfefbhedaedghfjfagicaeegjeihg 
#agjbaeiiifiiahhbbhbfecigibghjcgdgabgdch
fceaefbifgadeaaidcihfgjbfhbdebfgigafhad 
#chiggbadaceiegifijedbebiajcedhheigcajgh
bccbfidadaahegibcefieidicgfihfjabfegehh 
#jcabgiihdfgidhdfgceacdfhjbbfecjbddfbggj
efihbddchfchgbigjjbihgehhhjiecdaabfgedb 
#cghijbfafdccajdjiahfjgjabbegagehafffjcd

eajjgijihehhhjhjhdbhfhdjgdjjacdjchchacb 
#jjjdjefggehcadjfaiibfeehdiahdeabghjfhgj
hcghaiaihbjbhabhdejbbfdbjhagcjbbdcedfja 
#jiedabcchhgaegeihbecgcgfhbeicgddbfahajh
bfaigdgahfejdfcdagdebhgcbdhbgbfcgffaehc 
#gfciccdbabghfbicehdddfahfggbbfejjcecjce
acajigadjbbfaadeibjbaacgdcbjifgacjeeied 
#ejeadgbjdfdiibgfjhbciajbccdgcbciebjadie
cjddfhjjaijhjhiajajcgdabhjghdgeciceiafj 
#faafdajgihfbbaibiaffefbdaegbhfigajaifcf
cfiddcgehcfcejjhdeebfdieicfjgcbgahccaeb 
#icaadijbdeidfaeijdedgfigeicajachijhdcii
geaiccacccdhdccdggbjjfifbheidhgdcebheef 
#bigciafeaejgegdjihgbhchaiaecheejegcfacc
ahjbhfabdjeiahfijicgchgeijiefjddbgabeaa 
#hfcbbjecjigcagihcfdjcafjdbjjigdiaddbhdb
eeebbgcfiiagigbjiigegefcghbdeefjjhjgcbe 
#bdgigfgbadehdiheagajcgbadjddjffgdhgcaeg
ggbabjciidfigdfgfjigafhbegajaggdbgjbjia 
#igffbecaiehbfjcicjibbbidajbgbfgbdeddgaa
cdddgecejfafbbhhggfiaidgfafhffiejgajhhg 
#aafidbiidfgccijeafcdafdcefgcgifaafjghaj
ghbjhjibecjicdcafjfggcjhdcefdiejebhbbgg 
#adchiigiheiageeigbiedjdfcdbjiefgbbdecfi
eahiegcjhegieihfdcacdgicchehabdaeceefge 

#biahiiaegebbeccgbjaifibfiedhhjbgfdfjegd
jgfigafecdbihhddcagfcdfcgacbjchhjijbcch 
#ahiefdijbcbbdchdjdediibddgdcghehhieiijf
bejdiiijabeghbejhjfjjadghdbgcbaijhabhag 
#gaaicdajhjbdhfhhidebaddghhjfdjjhebiaade
ehjbahegbeijbciajhibibcdiejhcdefjcheeih 
#bfddacjejjjgjhedgicfbjdfdiihfbefcejaegh
jfahjddjedaghgdbdhacebjjjfigchdhcafeijj 
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ijfiifjgdahcchgddfijfbbfbdeeccabcjgjjjc 
#gdfgfaiiiibdhbaifeceibjjbibihefbccgfdfh
chgcbiiidjdghiciccejicbahhhegijgieehajh 
#ecbajfgdbbhjfffgfahecbfcfadbefdiehceaja
gjbjbeachahahbabfhbfdgghdbjjighdagjbbgb 
#jgdicddiigeghebbccdbjbibiejccjgbiegicfb
gjfbffhfdjcjadhgdbjafgcehajefaaaeaaiadj 
#iccaejbfdaabdececjchfcfahabhfddbjaehafj
jefibijhcefjfdigbfegiaihbbigfbhadghjfba 
#fbfibdahiejfjafiiiifffdfebcfegbeebfghbj
cfjaaeddabjaijjiabhgfdgdcfeeaecdgbedjfh 
#iggabfhbbejdgcdceaajdbchdejdbjhddjbahhe
jahfjidhbbacdgdieeicggaedaaihfgighfidjf 
#ijacfhfafiiaifabijbfjjhaebgighaaacdeeic
cijaafccecijccbjfcjcfbciibfjejicjccgeaj 
#dhgchghbeabicbcahbhfgeegbiaejafcehjicej

jedhhcbbbfgefjhbabgbjcgbgjbahcgbajfcfah 
#gffjcbddhcbjhfjjeheajhgcigbbeidbjdjbjba
gcbhcibjjicjgjbfffehajbfgehchjifhcaighb 
#gajadjffihfecbafeefeaijhhgfhecejafijbjd
ejafdfjeedbefifbgdiigbjaheajfaiidcibdhd 
#gafcfhhbgbhabeeicjfeaaajcegjaaijcggbchd
ffdadiagiejbfdggffcbdcddijcaajfjdicceig 
#cficjggeaedcficiafcgejfifcfhcgijhghhhda
ageahgfahedegiehibcbfdddieigejffbjiejji 
#ccgggajgfebgjebidhahgejadhgiedfahfbeeaj
igagicibhaijcgefcgcjfcddfaecbgddcghhfdd 
#bfjifeggaijcgihbhabbheigehjcdhdgjchjagh
efgcdddfabjedhjbafaegaefhafiegefegahgjd 
#fjdhbbbgjfbhfbhbaaadabfecdfcjeecciedcjj
beejfbibichggeiebhfdjhhejjdigifechhgdji 
#igfjehgfahdgigbjeahhadjgddiibdjijicihca
didfgfcjbgjgedgdebjfajbcfhcbiibffabbcch 
#jcffebjdgiejececieehgbfaedbgcibjgggcfcb
jebgbcdgedjagedhhagjaehfgadefihidjahiba 
#eaiaichhjcbideghfbffgbjaeccebgibffbiibd
cehehgccgchbeeggecegeiidahiggfiibfiijcf 
#ibiheccdbhdchcbgdeeidejcebihagjhiihjfeg
jgjdcahhbfebjaaefgjicgajddchffddcjifcbd 
#deheadhjfahhafgejjhihhiiigjebehbbgghiic
hdbhaahgeeifaagifihgjbagiiagjcefffbjjec 

#hhaibjicfibdihccjddiiahigbihadjafdgfdbc
fgbghabehhcgbjhigjbcbddcjafaibiidihgahi 
#ijffeghfgddacdhcjhfiaefiibcbcigeaaaggje
ggcjbdbafdahiaedichacaeaccahgbbbbajdhgf 
#jahfcjddidcfhgebecbdidcceacebeddihcgdhc
gjdeaddidjbiahaddihghafhjfaceigjabibceb 
#ejdaiahiacacdehgibcfefjaffjedhcdbhajfba
dfihjachdhbdjijhajffihdfagjgfgjhefhehii 
#gheefdhegibdbhbdjbaidibhfijdhbaddcbcbbi
daeabcdjbcdjdhgjfgcecdhhjhffagibahbjfih 
#bfjahggefefggheieebigbcfbjbjgbhdhaadbfj
hajafehadbeaagbceffcjhhgcdfffgbaddjegee 
#jfdgeafcbieagjibcgfifhhhdccgajdfaheafbf
ibbccfiigdcjihiajhagebibijabgddefgbijbf 
#higcihbfgcbbdabbjedhdcjgijacdihhfejiaed
ebbhjejedidiedbcbefcebafaijeechdcgjdedg 
#digfehjdfggdidccedigfhhffffegdigeififjd
fegicfeeifcdcehbfjcgejadcbeicfecdjadajh 
#ecfecjidjiccjgajbcfjcgjchajbajifjcccejd
dahjccddfejjjbfhbhagbebbfeffbibahgccjfh 
#ajccjhfjdahgjbhbeejdcjcaabhjhehjdhgefcf
fchgbfihgceccddfdaejjhidbhecihehcdfhdff 
#egbibcejadbecefdjeedjjgdebiagiiaehcihgf
cbaaddefhgfgfgjbagaihidfbaacbgcfgdccfei 
#babbaegifedgggihdjacejaciciehaegejbiiih

cjighhjhefhdchgghefijhgdcaichibbehcagac 
#djgcgeidcbdfiihajdgbfhicideegdihciggdai
jdhhagejegaiddjchaaiddhbgbfcfejeedafcid 
#ebichejbcigfcahdbehbhdgcgbfagheadcgcaef
ggjcfifjaijfdhcdhbafjhdefebddghfbabgdbj 
#hadfdfacadfhfejbhbfbeibeejcgcgechcjahjd
jafbjgaiidijhajajfdgeaibbhebchjccibbgfg 
#hfcghjebdbdihhcbeaejcccjajbbdfegaaeaigh
fibfebgjhhchfhicedgaafejejdheecfgjacjdh 
#jhdecgfabjfcahahhdhdidahjfijfjgbaejfiac
bifhhjfdecfagjjaiijgcdfdbjidfeaibijjjbd 
#cgcachhdaihbfejefjgibgadfdhajbfjieeejga
ddfgfajagfccefhceeeiaffacaaicfefgcgiiih 
#dcfjiciagbiifbbcbdehicbdbfgiceefiafdgfj
dhaeaijibafbhfgdfcedgigjhffddaiedfaeibc 
#hbidjaafbagicgafdicahifhhajejeaijfjbjhe
ciaejfaahbdbfigfefihcgcejabjiebjjigahjf 
#jadcgchghabfdjgcacefedabbicdgdeehdigjdh
acefaehfajcfadadacahdcffgdefbghcjcjecgg 
#aigaijeefegfjffbcfaicdffbaiffjcjciccfaa
degjcbhhjeeejcaijaddhehdffaeeghghgiegaf 
#iijihifcfgdgccchicjfcdgcdcjgjfejajcfgfa
iddcdaaifgehaehdfjgcgbffebjgabihdbdgbhi 
#cbbeehdhcehabcijbbjgecbjadhdcdjdgcdaedc
fefbccdacjachgachgjaieidiiaeddggcfjijde 

#eehbddgejecfjiachfiihhhcdgejageaghfjfeh
icagcabbcjfhbijfhjhaifdcbdjaccbgaifjidd 
#gcjcahiedbfdefbdfeccgiicedabhhgdciadjdd
fhebfehfifhdhhhjacbhgjhdjaddahcggggfica 
#gaiedibcaihcafiaiiihbhdchhiaabebjhjheaj
cdagiefibfbccfdjdcihaihcifcabhafjbhaddd 
#fjjhigfbagjjfiibdegagdagdbjajgffdceheja
bgdagdahdggaegbffaaidgdhcaaehdhchhfiaif 
#gjcaddcichbcjhicdbejhfghgfdiggaebdjaadj
bbjggfjheddfaijdefdadddfhbbdccghigjdjic 
#bibjcahfjcfjaachbifidafffdbdghgagbjehhb
jbdbbbeghhjaadeebcadaecffhcffibbicaaebi 
#gcdghfbagcddfedagigjbfafhcfhfdhfcccidii
gacjgbhfbijjegcgcdeaegafecdejjfffejaifb 
#ceaeebcbidegdjcgfibcdfdfjjedieabbjgbiia
djfaheihhieehabcjehdahahgdifeeejigaahab 
#afdedcbhgagagabgagfcibdidjagdbdgfgaeaed
edebfdbchfbjahedegaedfbjigchdhjjgebafej 
#fbhcebicdbjafbgfjhgfadgfichheacjibccchi
bghiedigfgdiaifedaejaejjhjjbhdhebabjhbf 
#gjahhdjabdaifiafcdidfadjeaichadgijiigag
baachddeajbdijaiadefcaeaagajiafcaahifad 
#bbbbdhjebjgbchhfeefgfdbbjddafihcciafiie
deijejfihbjjjeihdhhjhhbeccbjebcjbbahecd 
#chbajcffcaecjdadcdehgcjafjigdiiabicehcd

ccjegahihcfgjfcihfjbcigfegjbaghfjhghdad 
#ciigaeijhcdediiahdjbggghihcaaacijdeigfh
gaddafgdahdffdbffjagbgdacajjafhgedihbig 
#dajjffjidaebijfecfdchidbhgebhcgdfgdabfa
cgaaeefhabdebcbceeihhhcigcefddbihcbjibg 
#bfafcjdajhhgahagabddegahcjhibbcefecjged
ijjbfdjfcihbecbahdcbdahiccgggjfhhadfacf 
#dfihgcbggcccgedahfcabbacjdffjchgbfcgiae
fejcijcibdeicibagaggiicggedefjhdcjdbeed 
#afgedjccghcdcfhhiiffccfffjehigjjebibegi
ficgcjgfeacjcbhefafadbaaffceaegibedcbfe 
#gdcbdefdfdefcjabccfjadiheedihficahjjjcc
iiafedcciiabjbeifjbfjdjbcdjfcdcdcjfefdd 
#aaggabbccehjbejfagggjdcfbciifgehjfdjfcf
ebdcaeahjhebffhadjhhdehejffghaidaiegcjg 
#jbhegdbgjdabcbfdaffjbehadidfbjahibafhed
bcieejgdfacbgecaeedehcegijdfdgifhhedfah 
#dhbjdfibdbhdaejecibijihccihifaijhahhagd
gjibjfacdieehhbgdgghgffgdcehggedhgfdaga 
#decfjcgbfjfahcihabighfifedcbjcdjghjbehi
djhdgddgdebfcecbfgifibeiidjegdgdihbijed 
#ceejjacfheibjeegdgciaahhdbfihjccgbdiiea
hdjifgfebbbbcabjgdfchjcbigdgccdficbbhff 
#egagecijccecbcafcbbjaecgadedeejgfbhjeaj
hafcgfadijceieigbjiedjgcjicjfgbicjgfdca 

#ajecifiehgahdhfdhfahadgggchajbbfcdbfbhf
dbghijefaaafidjjhbgecaahedjchgeejfhfgge 
#ajdieehfhachgjchagedaiacbgfbcfeaciijhjg
bdgdeeidaiebaceebiibegfgaaddegfbjcgdcdg 
#djefebdcejdeghfgadeeebciaacjhagfajcdcae
ggafjebbjficjbgchacgbjfeihdaebgdcbfbcdf 
#jecgeehdcjfbiccaadfbjeehhfddddbidbaeffa
dcacabjcfgefbdgacfecaccgibcbdggafeeifac 
#caibgjaafifcijicaeehgjgjedebjihbdbbfcac
ijdhbcjjdbdcbagchadcedhgefbefgfdcddihac 
#cagefadhicffbgceghijaaffghjicjcedgdgcjb
chjbddfefibjbicifigbgeafaaebggbagdhbgcc 
#haiebdddcahedfcfijjdiiebgfdjbaehcaffieh
gaciiffcgedhbcdcehhgcbihgijcjdddbihjcji 
#ifgiajhfijffchgfccfhahcieecfgdehjcifbib
gcabbffcfaiddhhbgdegeabeieacajgjighiifa 
#edfhhhbbcfeggdiaahacggbffgcefbhhfaddedd
dhhbhiefdghbiaejhbiihhdjfeiedfdfjcfiiih 
#ghgdjiccaachcjjibcijdicfdbdfdhgcfgiacjg
ecahgcaadhdcbejedejijjhfjbdhjffeiiaghff 
#ggbchgdfehhbebggacfcjcghdgcajfgebhfidbe
bjgihcbihcccijdcffagecjcecaehieiccfahec 
#bcaiafaedghcbgdcccajdegchjgjcjgjhacagdg
ieidbhijehafgbcidbffffeichjdidfcbgfaaeg 
#hfejidihggcaeejbhjcjachgjcecfcjiijcbgbj

bhgchhdcgcdhejehbjacieedjafhdefjgfebccf 
#feddafeicdcccaabjajeiafcgibbcjbfbghadgf
hcigihhcbdgabcgcdhecjhacegabegccfbehjeh 
#diajdecbccdffgdaabjahcccjfccdjdbcidjhhc
dhhajcdbiddefahggebbgaihcebieebcggghcbj 
#iejbijcjidddhgfiijajbcehbhacbidhjffffee
bdcebccjfchafchejijjdjjbfcgbgcddfbjghdj 
#jehddafhahcjedcjcfcgcgdagicacdgigbaadjf
bdifhjdchahgifejghajegfdhgjhjijgdfjjecc 
#adedjiegdeajjabiabbbhdaihbcjhjjbiceechh
fgeagjjdaffbieggbgbggbabjfgdjgbibehdhid 
#iieefgjffiaghfibjicfhjagefbgafeidchbcig
jgiaejdachheghaefcfbaaiiebbdafdighieefj 
#gcaggdgceigedijdiebffeefcebejajedcjhgda
begcdhidihgahhbhhcchgjfgibdbedehebcijic 
#fhehdaddiffiegifjaiigdcjiecgaichfdieebi
bfjgaibaabjjcagjghhifiahgfjehdecjfcjdfj 
#fgeejagddcjghaebibaddbfhhighcjhfegfccdg
bbihbeijbafhdeeefddibfeefjgjiiaahbbdcjj 
#baedcdacdhcaaaffgahaidagfjiecijcjaichei
bccchjjfebbfheibajeejehbihhddhfjabgfibh 
#adgcgabehgajdbihiiccjighbgcgadgcedhbgic
hdgjeccahbecgdcddejbbaihefhificjdbffdig 
#hgfhhjehbjbdebjghjaafdddihbibfcdafhefag
fijgbcabefchibdagcdfjaijagbdhgeafcbjjjf 

#fcciajaaaeceebecfadiijebafebcbbahihcahj
bbecbdicebdgdabfajbfjeahdfbaibfhjeejdfc 
#jdcagjdghigbbifhgchai   quit  bytes    
used#bdhchh1 alloc#bfbjif1 time#cd4eg   
----------------------------------------
-------------------------------------   
exp7,pi7 to #ejjj digits4               
#bc4adjfibfcbggibfijjegbijhfcfgidhedgchj
bffajfbdbfjjbaaiicddejdfdjiebdcdbcejfij 
#debghceafigaiigjfgedibaifgeiebgjdhjajhg
ggcadddbhjdddadfichcehddgagddehgifjihdi 
#cfecbgifehfcffibdbbcjbfhiiajacgdagfdfhd
djadajcieahchfhdggbdcjfhjeiehhafbddihdd 
#diadcjiffgghdacfgafcaifcdadghdjchbafeaa
bhgfcggcadgjcdgcechccafbhbabidjdghiaicf 
#bbdhbjbbajjfjcbjfeddccfbgcfeegbgahbcgdd
ihifahhehjeieiafhdhgbfdedgijaccighcdjbf 
#eieajadiifdcgibdbbifhafjgiiefechadbcecf
bjfiegfjjggjeijdfjhiihhcjjbbedcjdhgabaa 
#giacjjhdicbgcgiehjgbidbgcjaicajdbfjafia
icicbehecbjcdbhifhffahiebhcbijebagaaaeg 
#aheaheejfhjbbedaigbjdeffcgjhfeefhchfhcj
eddgiibghagjdjgdiggfhedjcfgeefecdiegbah 
#hfghhbefciidchdgahbbdehehhidbhecebdgbfj
efhbajbgajgfjahdiaecdeahdfhjfdhhgchfggd 

#dcifcjeadjjeafiddjedjffebfedcjifhhfijfc
icgcaeceihcgcaajdbagddccjbcifghiffijjce 
#jdaahadhfjeccijbhgbjcgeiiahehfhhfhigdhg
cbdcbagbaeheeifjgdccdfgfdbfafghefaagcec 
#ccfdbabfefcaiaeffededhibbhifibbdeggchhi
egjijecfahjchcfaigeajcbfefgidcfbdjhhcei 
#ijfdbbcdgabhdfeccdcgcadhgagjfeaghidfcgd
bgcdabfidgifhjdcbajdadgfffhbcjbhfdbicdd 
#dfghgdehcjcfbahifcebcibacbicbdcgeebjbcc
abfcdghbddadgejhjjfbghcabhbibgaeaicfeaf 
#fhcjibifbebbfhbigaeebabegdajichbhfccjjc
daahigadbejgjciafahhjjaeeejejchjceechdc 
#gahhfafjcbhhhhfaddbejhiihhajjcaigddhcci
abbcjcggbebicfagbgcjgegdacdcfbgeijddbfi 
#iedjfbhibbegijhgeggjebiaajadhgfcjbfjajb
jebdghjjdbjfbggbceacigaiejghcjddhhgfbch 
#cihafcidaagdddfhbgigdgcjhjbdhfehhedcfif
bgaifbgbchhjedffaheecdaigbicgcfcgcfhdai 
#adhfiajbgjcecbhaecgbdchfgdhgeeaagbadbeh
cajhdgdajafibbecihdcdjchihjhecdiagdjchd 
#dejgbaeaadjhgfababehaaijhcbaccbfdibchce
eicdiagedafgifbbhbcbbaiabbhjaaaecbghaee 
#bjdbaijahjbeijhhiagfechciecfgdejbhijijb
bdgigjdjifibdjeghejhbgeajcfiacefjceddbj 
#hgagjfejhfbjdjgajihjgfjdgfdiffddihcgcaf

afcfjcdejhbgicgjafecdjddjigddafdbjbcgjc 
#dajhjfabbhfaececjeachgdfahcbgdjaaecjdeb
hjajgagjijajdijbaffbafcecggfdfdiijccbfd 
#icbbhjffbghgjbhgfaacbbbcbfhhjccdbgcidgi
gehbighcjdbhagaidjfhifhcbfahfiahigibbjd 
#cgbcdfiebfgiedbeajghaegaficcjcgeabhfjih
dgdgaadegfdiiichebicieecagbiicahebbdjcf 
#ceheafffdcbcdfjbjffiagechicfhfiabdbbejg
jccebjfgccehfccedgcehhgehbefhejgdaechdb 
#jccjidcjhhicacacdegjegafjccjhhchaechhjj
ejddebijghfcgjcihaeihagaigjibagagijbeaa 
#hhgfifgbdjbhfcbhdagjcfbefgbdidgiafeacbj
bchhieaedeifjhhdjhjejahfbijhchehjdhajce 
#hdcjheahieidcehecbafhafacifciceaghjgded
jfcddfcciifhgdifejagcjegggggddbehaafece 
#dfdbeggcjggadbjjbidedabcdjdghghdgcafbdh
dffgdeaebcfajejhhbijegjcfjddihbcjcgfgjc 
#cdjbdjbbfejdbcechidfeijjbbcjiighccdgfdi
fadiajcefehbbijcefeejgcagffhecigeiaaiaj 
#gdfbjbfhhijhchhgdeacefaedjcbcibhhjbbbba
gbedeeibjeabfiajdcigddgcjacgiacbafdchfb 
#jdgjaaeaadjaiebbebccadeffafjiegcjaaebge
aihajegidibachijegehahiifjdjfieifcabjah 
#djjcdhahaafidicijjejjfcbgfbibcdgebdcbgg
ibaichcbhbafchejbabgjafdbfcfhaejbhifigj 

#fibcigaahfgeiadbjdcdfjbebjiefgcgjehgdhd
icifhfdhagcfhcecjihijhhfaghgegjeecejjih 
#aijbccchbeiiihjdeegbghebdcdciibgabhccje
fhgcfdgjhfciggahicfbagfjajjabdagebiggje 
#ffdadagibagicfdheefggcifijhefcehidfedag
cdhdijgfhifiibcefhffafdjeahbbdhhdfjcjid 
#gijefieiajbbdiehffjabjaaddefjejachgjicj
dhijibchdaaddgbcffjchiegejgfcffhhjafbdj 
#ibghgdchbcgcgddhadiaffejfadifddaghcedii
fijddjeghceibbgbddicabcfeaeahegffhjfjbj 
#hfieejbecgffddffidjhbbeahceabjggghibeii
jjacbgbbbiegaaicfahbiadcecjhdfdjbjiadae 
#gbfcdefibbaceagegadjahbbadfbjejbhagjdef
jcdcgadjcfhgbfiaehegfcdjedegcihbijihceb 
#bhcjggjbibhgffgjicbbejjdbfciacjijjejgci
fhbhdggjcggiheffbedafhdbgibdcbeejbjcdjb 
#dacdbhedabidghiijijjjhihjihficdfahehahf
fhhfjgjfjgfbjijiecejjccidfdfjfeafgcdeja 
#haadhcfbhbdbbebehefcbhjhgcafgcbefbegdcd
eihecbgfgdhfbgjgiehbiaicdieaijecijcfjhh 
#geidaacficahbbcdggggcafafhdebefhjdhaaeg
jaefehjhaaicfjjbgcefbbdiahafegicajgfegd 
#jdigfbdgaccgadiaecdhhbhejeecgdihddjbiga
hjgbfecbdeihffaebghahbdabgfebdiciiiadeb 
#bicgdecdffgjgdcgbebcacdahgfdhddefciebbe

fihbaegbdidafcjcfbaaigfgaidefjachcbfdjd 
#jfgdecedcfdedejagggbibiggdjceachfhjjfid
eejgggfhdegacacbhbidaadedfbdeffaceeggea 
#djciffjjdgaeaiaeccbcfgigajfheicfgfeccac
jhggjeaeehbdiffehiaedccefjebgedajcfecjj 
#faadbfabicfjijdddbieddhfahbaeiddcaehfci
efhbhffhigebjbddjibaecabhjbajhehcgbhccf 
#iecdifhhbjcagadgbhacedeaaidfbeciabcagcj
bfhaadbjiifghagjgciajdiehddgggiacgjfjba 
#bicaaeecjdhfbhacjachbbfffbebbidehafbgah
defjbhdfchfehgijjddhcbjbbcjdjfhcafhaiif 
#cafjciecfdbhaefgehebjdhbfijajcihbgdbjbi
dedihjjdgijcjcdadfadiibagicchccfcfghbce 
#ihabbajfjhicecdgahjhcijahigbbhghihfechd
efjefifjhjghhgjcbjfgghifcbdijfhfgfcihbc 
#gceegdjcecgeeehdhaafjhjihchfaibgjeeafhh
ibceadjdfiijffjeaaadjjgbhifaccgdbbjhfee 
#bcdigaahidbdibbhcdbcjadcbbdjfhaehhefiid
hadhbchhgiieacebccgecbbiigcddgheejcbgfj 
#bebhhdggcicbfbajdffhbagdbficchicagfabie
bejihdgdgejdachafecefchdjfbfjhccdbijdbb 
#dcdbbjijjabhabgfecibcbjeibeiebdffgfbbeh
eabbdafaicaheefihigijgddhbfdjchjbciifej 
#ebfdbdidabhjigddgbfbdjfjbefiggdfcddacjd
jecffeajidebdbfbhhgeachccjijhbicfacdfbd 

#gdaiccbcgibbiegcjbciidjfeeaeijehjbcfgff
bjcijcheieeghbbbhdhfhdjjcadgbjfcgfcgfhj 
#haahgabgheaaadbedfhfejjfgedgfhhicjehfai
cbjbiafggfjhdgegahcgggajgddgcgegdiidhbb 
#debceiccbjgdfgjaeachffhjfdfffecebagefbj
chddhjbbihfdajejhbhhjfhhjdcchiccecjgffa 
#biifgggbhcgicejjggebdheehgfghiaidabhcie
hhcbbbgjieaddcagejegfgdhhhdcieabcceifhj 
#hfjaechiihafabjcdchfagjafdjjjiidiedbhbf
hfbbicfbbbdaiafjbiiahahfefefjagiabgciia 
#daghbgbic                              
----------------------------------------
-------------------------------------   
exp7-,pi_/#b7 also i9999i to #bjjj      
digits4                                 
4#bjghgiegfcejgfaijhedfieefaihcdighggjjc
chgghdafcagfifjhjgeacehhcjeedaihggbhedh 
#bacighhfjjbgghfedbfjcecdjebaggccjgbcejb
ahjhaijfaigcjcgdffcihfiiiiaabfcaghfdabj 
#egcagagggiebjjfgdccgffdiedbbdfchaibigcg
dcjechgjcgfjjeahijffcjccjdigjjeaijjeeef 
#bjjdgehffbjebidceahcddcahdcdeejbgdgigde
cddgfiicdgadagbchcbcjhaebgadhahjjgfjiba 
#jgdaibjdgaeahghcecdhiehdhbahijahfjbiehb
ccabieffbiebjgjhbcdjiefgfifcfcgdbjcidea 

#dcicidahchfaijajhjhbjhigggageagjejjdcdh
agfdedgeagadebihidcdaacdadbjagefbbaedhh 
#jiedaiibjiadgceheaebhefgiecdebfigfcjdii
cgbieggbidhbeiigjbhdggebdjcbdhjhbjgggjb 
#iahgaigbagechcdgebjhfjhfdhehgecdgghfeed
fihchcbeecfgijachceagbbaahfdaeaieieiabj 
#cijdddhjbbffifgcfgidceifejbjeehdcfjbief
ifjfeehbdidcaccfidjagbiebdbhifajhfafaih 
#bdiiijdeacefijjegcfdjeaajbffdciacgceagd
jfbgijgdifhhdijabbgeegaiaggfbjcggcjcehd 
#ebhggegegfjcdiejchabiiaecihfehgcgfeceia
fhfdjjeaeiihhigajfcgiijfafjhfcjjcjiijac 
#fdegjidihachadchjcffdjcdhiacdefbhgegafg
giibfccggjjjgdiehicdddbcihjbibjicbfhbcj 
#fcbebdighefafificdijhcdjbeidgbdhdggafhj
idfeecedgfiafhfjjeebaebajagbaeafhbifaae 
#eaecgcgbjdjdbcjggdgeedbhgfbbeecfjaeabig
jidhjfhiggfafbdfcbicfdhgcbbiieegiidfbbc 
#ffgefecfdgeiijaghhbdbdjhgahieaddcfgbcie
jaddddcbjcgdjfijjjddheadefajbifiaajdiaj 
#hghjegbebcegfjiabbdagghiahjiccdehddhffi
hijgeaajbdaaeifbebaaihcjhchcejfdddiiiic 
#ggcjadbhcjdahcgfafbcbjaeeeabhgdhfbgbhdc
cjcbehjgidahahjhgchfaaegagafadhhfgahcch 
#gcaiaabibdecahahcjgejjhfbcdeiahhgfcfhbh

iddccaeajdbacgdgdgcbgihjejccbhgiegbehfg 
#edahhjdbgidbdfjjhbeeehicajeiaeiigjjjcie
fejfbeddacfdehhbffdffjcaabgjfgdbbcajcje 
#eceahgijbecachhigddjegabagihefajdjjffac
geiehcbagffeedchajjffdcagbcgbedgbifdiia 
#ifahefbibhabiiiecidcibceggahjigffjfbdib
ffcaiadegijjaajghgdfehjihddbhjdjhdeahjd 
#jcbghjfgbfahjdbjdabfdjfeifbgghijfgdjihi
dbddfjijbdedhcfgadaaggbaedihhdiefidjdeb 
#aabdheiabghgcbaacdjfagffchifcfbihchfgee
dbjbefehghgbjafieaaiegjhcfheeihbghfdbgf 
#fhjgahhhabcaiadbddciahaabagfebjbcejjgfj
hifefdcdhfcfgeicjhcdgjfcafihdccifcdbgfb 
#chjhjajjbdbghbcbhcahggaebbbbecfdaeiebjf
hghaefbdhcefbfbegadfjcadffehjibcbfhdijc 
#egdfigjejaehifghjbcebjbeihijfcfbbihbahj
hejcddjfdgei                            
----------------------------------------
-------------------------------------   
exp7,pi_/#d7 to #bjjj digits4           
#b4aicbhjjejgchjaedefeeigfifeibghgchbbcd
fafcgfdaiidbgbccdhehjaeiahfegjbfhfdahib 
#cficdabfebbhabeghafidjdgaafggeicegijgfa
efidfdgjdafjjhejgfbfjecggcbadihhbegffcg 
#ahjebigegachjdefeighgbgbdggfgfgddgfahge

ecabieiceebecjefefabaidaichcbeggaiccjgd 
#hjfjhecacfhbachifagefeihgffeajiffdcfeec
jiaeabjegjdeceffjihcfecifafeeccdfdeijfd 
#cgcejbjgfdghcfjdieiacfagcedbaebighbccjd
cghjjbbhhbifbcjehjbcbfcghfjghdbccgdbgde 
#gcegjbjbbhheaiihjijfjfjeeagbgccidajadib
hfdadjafihbifeagecebbcfhfejaahagiffgfdg 
#degfbfgahfjajbdfahcedjcaadjahebhchggieg
fddbdaihgbaaecahfidjcahacfaafdcbhjiebcj 
#gcjjbdcihjdjgjeffbbbfchidfidcegaifaaddd
jagjiffchhceficebcbecdbhgfefhbfabbcdeie 
#ieagjhihfcijgebdjgdacjbhgfffjcegeiijhgf
ieddgahcffgjabbddghjbeegcbjjaaafeibjdhe 
#gcgdhgddfdcjfichhgaagffgaffehgdbcedggfb
hbbeejagigjdeeahbbgegjacfdaggbcigdhdafi 
#icdbiddbhidajbbcdcfjdihgdhdgjgccgagghce
dahifjfcdfeiaaajjdddcehcajfafabacifbgcb 
#hfieegjeijcebgecfhegdbchfhcffcjhagbbjcj
abhcabebicfbageedeiahgafgediicgigidgfbb 
#abiagcibeegeicfabbjcbieehbagejgciaddgij
ifgdhecddfjecegffdbbecfdbiiechgfecbdacf 
#idijahdhbggdbibfidhdchbbecjbfbbbfhfacgb
ibahjbehdhfcidjgfjaefajihahcjichdegjjfi 
#dgjgiefdjbjihidaijcdecdihcghaijgfejjjja
jjfaafhdgafejihaiaaajgjchahciggbchdgfcf 

#jhffdhjefchhbgeiiabeeeejdbfhcecebjfgdcj
fddjbdaaghfigdifhjhefehehgcegaagjgdcfeg 
#acfghfjddcaecidgejjfdegfahjgheadgbedhhe
iegieaccfbgaffebhcgeifibeacbggbdcabbidf 
#eageejjdhefeahgigcefhbedjadcijbdhdhjdfj
jfdjfaghhhdcggcggdejhfhiddcfbiiadfhijgj 
#hfcigdcbjiahdbijacfbhdbbfejcajeddbibbee
didfifeahhdbhabbbejgibebagejjgbciibeeea 
#jbiehbfcejcijibdhfbgccebdhfiabgihcjiiaa
jfcgcdcjgggieihcaffhddbdiihecigieedhabe 
#bfhjcciigacjjaadiihdfjdbdhcfigjaehdddad
jadhdaihbcjgdfjjbjdhiecgahcjfhdjgaeicji 
#eagihcbbecbjicfcfcfchehaefefccdgdiaajda
heiaidccheababeceacahbgjaiffbgdchfjeeah 
#cjdeaeghaghecfhdgbebbehgcahcghgeaaedbib
gaafibihegejfdafadccggeigahbhaagcbidjfd 
#giaggehdhjjehifcjjeffdibcjihhdcaaagcdci
hegiccfjdfcgeijjaagebcfdbdghjibejjbdaca 
#ffhbjdgedgagfacbcadhegieigjdbeheajdfjgj
gicbcbdiicgacfdddcedggbgefjcacbdecegedf 
#ijghhfahfjfdejgfdhbjhfhbfjjgcgfabcccbjj
ccihbbbfhhii                            
----------------------------------------
-------------------------------------   
exp7,pi7-,pi to #bjjj digits4           

#ai4iiijiiigiahidgegfgbffddbihdffijdddif
jfhicfhdcbbeajfagbdgjajahagbafebeidddjd 
#bdcghdhhhicgagagbedcbaeafichjdfahbhghje
dffdigccdaiihjeadcbecfabiibjhfdgadhacfh 
#bdghggfhagfjifgcjcgjiafcdcacfiaahhaegbi
dgajbhdcjgeejegejaeggacdfacdeifhfhjafaj 
#gjdfdghjaejgbaagfbdhfcadhdghfjegghfgijj
hcccaajhcbefiecgdfegbiacfhjjbjcbccjdibi 
#fahejdfcbhcaaejeddebbciiijgcjcahjajhchj
fbagbfbfgfiiehjcedcdbjiffehedfhgfddihgi 
#fdcaeiihhjcdceicfefiggiejciigcdbhccacej
jhiegeffhaehgigceeghaccdibggiaibdijhdfb 
#bbcidhifcegdfedfhiejadhiaahiaijicdgaheh
bfeifcdabedfghehhbfdjejjccfhiebifigciff 
#dhcjjefdeheheeadbfffecdiaidegbciafcddde
hfiihjhajejajjbcfegfgigbbdjdaabgacifcia 
#jhbaaabbabcfeieajijejidhgajgjgjffhjfcei
cdcbefhdjibidfhijfafcdfgfgegheaihabghej 
#higfajeeghicjdjdahegihjabcajabhjijeedcd
afbeddjdihgfgibccdifcjhcjbcifiebcgaaihe 
#jijabagedcbjegdaijhheafdhidabgdcaeejegi
jbafgiaiibggcdbgbifdifdaafdbcfffgidfcdc 
#cccbhcdbfhgijbijggibafhcijhbgafbheifbbj
dbcfjagfceejcdegfhgedheghcfghdjfabbddgg 
#eebfcdgeccgfejgeebeaecfhahdhiciebacigfa

bgadhdhahahefjhdaahbejefdfibibaiaagbadc 
#djbchhgaigegcecjbgcgjeciejacefadebahbha
jiajdeigaddegeejbgjjebcahfijceijbbadgda 
#ciifachcajjgajhaadcjifgaeijcighgghjcdgj
biejcedadfeaaejjhfghafccgiehffcchfadebd 
#dfchjabjbcfaahddgcaaahijjcbagiahdehiigg
eahgdjjfdhfibegjagabefcaaghgffefjjigejb 
#aaffacdddjjbebhihbighddciigddjjiefachid
jjegicgdeacfgheidahbfjdbdghhdabifggadhf 
#bcfajfigeaejchhghihbdecgbiabccbjfaacffj
fhdcbbdgfchdjgccedfcibbjdefgficfgddefbb 
#cffccfbdeciicahhabjjbgdeebhhcjecjhiibbh
jbdffddcefaceahfdheieeccdhfbfdijdhacchh 
#geigidcjjfaabeaijifbehfifffghhbgggbajgi
fccbeebgcejbdfcafbddjcefdcjbegehfijhcdg 
#gggfgfdijafccegjgccdghjibhjghcebeaahajh
ddghhhgfbfcbchifcihcceehhfcfifcdeehbjgj 
#gcjbfdecejeahehidhddfddgdabdadcafcfaegd
cadacejabcjdfacajiaciijjgfbhichgifjfhdj 
#hiahbbeeccgeigcbjeiaiebciiijjadfabgfaie
iacciiiaheaehhhefcaedhcjchijaibffbdadgh 
#hdfdifbghieijfjghddejgfiafghdidehiecaah
hjgfaggffbjcjdfgceegbbijbgfigcajeffcgae 
#gdedjacdhdcgeggbbegeicfajcbaagdddedahai
hebfgbbfbgjdidfgbibbegffbabjcfhbjhebaai 

#jgajdhhiiffehhfbddbgghfcjbghedbffdbffic
fdcjdbfdegfi                            
----------------------------------------
-------------------------------------   
exp7,pi7_/,pi9999,e to #aajj places4    
#a4jcjcdeebdbafbajhcbddaeebdcgeddadbciac
caafgdecedbfcejdggebjfjcgfidcfhehccccfg 
#jghdffecffcdbiifecahfedacgbaacdaabaehfa
dhecjibfgebhcjfgjhaghadadcaadhbagcggdcd 
#dfddiadgcececjegiabagjfdeheagaiebcghcab
eaeghiedhejiidffbccigdhhgjedaeghgcifeih 
#iadabhieffiecdgeecgebeabfcheejcahjhbgga
jiadbgjhcgfieifiajgjaebfejdfegajbfegjad 
#fibhfiejbeajfbchchdibjedifjeabiigggadgb
eeiaecdheahdjcgcbhdgfiiidgaacaajbdhbage 
#ajhgffgjedjecegeejfdajgefgcecfbjijfejgj
jfefabjhccgaedhgifjeahbdciffiidjhhfegac 
#jgjaaideceiaebbefcacjbfaejecgeegcghjcba
ddbbjfcaeaahdabfccgjahbhbjecibajhebeghb 
#dacaieccjabgbiefjffgaiigdbgajhjigeiaagi
hfjjhcdddbddibgddcejddafdgcegjdeggafgda 
#jbgcfaiddgijbgfbhehehijdcgfciaefajeedfj
eachddjefdhddhdghfdgcjeibhahgebhhgjeiii 
#fahbdbaaheafcddbjffcgdhfhhijgcccefgbghd
hjgfdjhaifeigicffbbfgidgcjahbfjidaghbhf 

#fbheefbihgahbicahafdjhgajahgidhhgajgbae
abjcghcehjdgijbcfhgcfafcggdfjjaacecfhhh 
#egaecjhjfaafdjfedfgfiieicgdfgjhbbhchhca
eiaiiebdfgciciggfjhbeeaibaijddehabjiahi 
#efcbiigdaafccccijabhebggibdibjadibedbej
aeeicjbgfgbaaehbceaahfbaidbjfjcchbiiced 
#cfefjggdccidagagedchbjfafcehcebgbdjecdh
icbidffgiicceifgeiejfbjfacjjaghbhdgedah 
#iahcjgade                              
----------------------------------------
-------------------------------------   
,feigenbaum reduction parameter         
#b4ejbijghgejiehibhbbbhcijbhgcbahbaeghfc
habgacgfgbgadiiggccfaibjef  ,feigenbaum 
bifurcation velocity constant           
#d4ffibjafjiajbiijfgahecbjchbjdffbjafagb
ehaheeggdgegfhfcbgdefeacdc              
#jjdacdccjbaacadgc   ,references3       
,briggs1 ,keith ,a precise calculation  
of the ,feigenbaum constants4 7,english7
 ,math4 ,comp4 #eg 7#aiia71 no4 #aie1   
#dce--dci4  ,briggs1 ,keith ,how to     
calculate the ,feigenbaum constants on  
your ,,pc4 ,austral4 ,math4 ,soc4 ,gaz4 
#af 7#aihi71 no4 #d1 #hi--ib4eh;,f#ad   

----------------------------------------
-------------------------------------   
,fransen-,robinson constant4            
#b4hjgggjbdbjbheaicfebbaejaahfeegggbicbc
jhjheibjicjaihbiabbjje  ref 3 ,math of  
,computation1 vol #cd #aihj pp #eec-eff 
,math of ,computation  vol #cg #aiha pp 
#bcc-bce                                
----------------------------------------
-------------------------------------   
#agjjjj digits of gamma1 as calculated  
from a value furnished by ,jon ,borwein4
 gamma or ,euler constant is            
,lim7n-infinity7 sum7#a_/k1k#a44n7 -    
log7n7                                  
4#eggbaeffdijaecbhfjfjfeabjijjhbdjbdcajd
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bdhjifjejdjaddhfedbhcfbbdagciigfddibcec 
#fbecejjcccgdbicgccggcgfgcidbgibeiebehbd
gjidiafjjhgcebjcidhafefgjhecbccaeaggffa 
#aebhfbaaiiejaejgihdgicgdejhegjegdjjbiib
acedghfadffidjbifjdcbedbaeaijehggeecebf 
#gccaciibedjabifgdbjeacgedaciediaaafheaj
bhjgihdbcdhggehgbjejchdcajiciigcfacgbee 

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ifedigbebjcfbabhgibbfeeeiecffifbhagfchh 
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chijfegbifgibifjajjijaeabeaieiejibbbdce 
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jbahbhddjifjecggbdcdbjcbhedghcfgjaeaggc 
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#eijieibbgjdbjhihifbgigabeecfcbagidhhgcg

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jahfbfibfggfdcgbffjegaagfhaccfeeijhhaee 

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#beefjhhhhgcebhjaijaiaeejhifhheedfhbeibd
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#fegjcafjfgiiibehgafffgebdgbaidjigggihja
hfcfbfbefccehbebfbgidbbcicbedhfjacbfice 

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bbbdhejeaiaeicfhbfijgaejjdcaeehihgahjbg 
#hccechdeddhcjiajgbdidihjeghhjifagagiifc

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ffbbfjigaihiehdcfgiidcjegbjcjadifchjejh 

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#bagbegcbijdgceiidcihhhcacibgiibgificigj

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#hcjdhcaeiaifdcjjbaddabeigjbbhijfcbjcagd
ajaadicffdhjiebijcjaagafccdecaiagibbici 

#bdbdbhggbhcghgbcdiefiibibcbacfjibbcdidg
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eaichbgbhbhebjiadgahbcgecdcfeebedjbjgdf 
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hfahaeaagejfhgjhhgfdfgafjigiibibjaggfif 
#bdifccjaegehbibeifahhdidhehgbjjbbibbdhj
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#iahfeabjjddibcaedihcidggcadgceghfhhigca
dbjdgcajidechadfdhbbfcbajbccajgidcieigd 
#fbhebcedabiegeaiajfceehgibcjjaiefahacab

idfabjcjaegegfcdjaeighefhageagdihdbcggh 
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#bdcbghjijbaebbefiebahddadebihfejhceeabi
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#aeiffdjjihcbjadebjjgjdhjceeefhehiigcjii
haejcajdaibbchejhigdecgcbgfabidgagabfha 
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#gfgacehgaaeecfecaggcagideeecadgafgachbf
iggjfebciefbegbcjajagjbifiabebfbfdhhaig 
#aiabccgegajchhiidhcifeecdbcagadabaiadgj
ajcddgebfjbghfefjigihbacegafibgdihhcjaf 
#cgjhhfbiggihjahefbfibbgjcdjedaihafchghh
jhbeihgheagheifjgbdejbibaiicfedfdcbhede 
#hdeecgjihcddjgidccjjaehbgcbddgbaecfhefj
gbfibddiiibddifefjhijeibdadhieffaeifhge 
#eajdgedcghgfcacaiffjdehhghhddeaeijabcac
dahcbieadhehebhhhgfjedeagiihfcjibcfeheh 
#bhdcgighdccfjdhcbiichgfdhghijdfjebhgjed
ajggdfeciacfjaebhjfihhbehiggeicjfifiabf 
#gdcbhhjgaebghbjfaidgdcdfejdedffijagcjbg
gidahccgbhaaeaedhffghgbgedfbaechgcgiaae 
#eejgbbadchjjfabbcjehbfcjhdadijhehfccahg
jeabbhfhcfgfegfbhgaaaebdacghbgahgibhceh 
#ahcaabjaigjfjhhfhhfifcdicijgcfghcafefeh
hhcbjbagcefhjicidciiiiheejdfhgiabfbbiii 
#ejadjcgajddfgieddeijbidgcbabggdehdfhcfd
jegeabbeaadgfbafhiegaheahbeabbicigacfhg 
#fijjcdifdjidahggfbggfefeiahghejeaaffcei

ggjiaafaajgggjijefhicjdbbcbehffcchhjdbc 
#ccbfdadgfcfigjachdffbchdagbajejjfaegfcd
ijagjdibigcbhedbgebiefedifabjdjaeaijdgb 
#cbdfjaiefgaaafbhfcgabhgichbbghicfdbjhjf
bbgbebciaddfcchegeadfiijffcegghccifjbfi 
#affifhjiedbcciiebbcfiegjjfedbddejbcbbfa
jifhddggciaihcdfdfbeeccfjihcfjajjfhejie 
#ggdifdeccfffbjhbigbbbaeigjgjgcfahdbcbja
dechieacjafjjhjgiabiaacdgdddjeicbjfgeea 
#gfcbiegcciehfejeefbciacchhcajdajjgcfaeh
ijdgagegdgefebiegcdccejeadjijchbfiiifii 
#fjcfjgjcibfaahhfbgibaibaaifhgdjejhjbbch
ibajdfdhcfjiffbjcbghjhbcidcgdeecbfigcch 
#fghccchgdifbfcibaeeabfihcgcicfecbfjgdea
baaeeffigahdhcegffcbjfhghiefdbbhbacffbb 
#gaehfgdaieafiaaafehcffbfhffgddjdedeejhc
abgddghabcidiajjiicbchhdgijjahiddchdjhj 
#hdhhgejejeagjefagigddaeiigabieegadacgej
hbbfccadjcbjffehdfbggbcidgacjfigjadcicg 
#ghibdfiehafhbgeffjhgdcfijefifaciaidfjad
fcbdehdbfajeffgjbigheibhfabcbgbdidgghje 
#bggbeiccbaiijbdggfghcbihbabedjbjeacbaaj
adaafdddheahdfjehdggigajfhhfeehdhbiajie 
#cahfbaadgbghadfdhbaaajedggaddaiafaiaagb
  bytes used#cjjajjd1 alloc#beeedcf1    

time#djbf4fe  quit bytes used#cjjaafd1  
alloc#beeedcf1 time#djbf4fe             
----------------------------------------
-------------------------------------   
,,gamma7#a_/c7 to #bef digits4          
#b4fghichecdgjggdgfccfeefibidjigdfggfdda
bhfhicggiegcjaajjiejdbhcbg              
#eijdagfajafggdchaiedjihbhhijdaahhghidai
aeijdibjjjgbbfcccegaijhde               
#fiejddgbbeiigggaccfggjhdfigfhafgbhihbcj
ejjjcbahcdbeejcbbbdgaefid               
#ahageeeddiiebgbhghdcidggiddacjegfehbhdj
afabcaiadaeifdffebfjccgbg               
----------------------------------------
-------------------------------------   
,,gamma7#a_/d7 to #eab digits1          
#c4fbefjiijhbbaijhcaaicjfheaeehfgfgbjjbi
ieafgfhbhhjjfedfgdcccggiii              
#efiiaibdcechgbiabafahcfjacfgbcchdcjjcfa
dgageacibdbjgaiifehiaebdj               
#idjbbeeiiggdbfdehhijcfadejfjfdacgddhifh
edaidiiibjaibfggcjcgiidfc               
#jhibbabdabcahcbcgjgiibjhdcigcfiijgjicij
efbjibibcbcdbhgjbgdaiaddh               
#fjciegacfhcejcfhfedhgiieifhcfhdgfdgehea

dhijijdjdaffcdjgfcjccigah               
#jffhjeieggcdbcgijheeijhjgadeghcabigfcef
cfhhbeehgibhhaaaijfceafha               
#ehejhdidgdhhaejbghhfgcajgcajebdhgihbeaf
fcffabhgicafdahddagddchbg               
#fdegedhjjiaiiadggfhjaibbhaejibfaaiidcbb
iighcghcecfcdeieedcdaidgd               
----------------------------------------
-------------------------------------   
,the ,euler constant squared to #bjjj   
digits4                                 
4#cccaggibchjggahfgdcahcgfacfceebddbbffe
idagadjbdifbigdcaejhcccchjjbbfegicfiege 
#fffiffabfcbfhfcagaeiggcjcjciefefjcdjbci
heiddebffiibfiieihcfeebgihccacghfjdfjcd 
#bfhghjcgceehcgaabhhcjigeijhedihhheebgea
ajfhbehjhbhjjgheebghgcgefjffhdbjjigdacj 
#fagehhfiaaghfidcaejejcbddhcgjbgfjffcdbb
bafiicdbagjceggaecbecfjjjfjgdajaebgfcci 
#iieieeidcchaegbhfgdcgeghcdbdcdbbdfeefhb
cicbdcdgjdjehejcbjacfjegghdfeeffbcjehjf 
#ccfeihgchejdefgjebjjjcjbjadghjcabhdeddg
ffebfbfbhbdedjijeicfhaaagicgahfijcfecia 
#ghgcgdfddacffbhbcbdghhbaegebjgdciaiejhc
jbcjjcehbebieibiaajdidafdjedighjdjgigce 

#ijfdadedajgadgcjahhccdbcgbhejegbfifdicf
ihggebbcghbfdjbjfgdddccgciieieebjdgejab 
#bhhebfbhgcjdcjefgbjhchjgiaghbjcaddhcccb
cggahacfgicaafjhdcdihhhafbabdibbhggiagj 
#dbjcfcahhddhacijjjehgghbcefjicedafehcjc
ijfbjjegjhceagcdedgbaceeceicaegdiggbdgh 
#efgegjhbdeejghhaghdfijbgecjhaeheegeccid
hijbbciibdcbahbfdfheabbbcgfeigiddjhbbej 
#feaeceiaagejgiebjihheajfgcedbfgbgjdbdaf
hjbbjhaigbcgahfgahjeacfbcfdahgbbeggfdgf 
#dgjfcgabiajibijeijdbbicgcafjcdieicajcaf
ccddghbeiafcfdebfjahdhbaacjcadijeacbbbh 
#dbdeaecejgahiagdcgjifgbighidcabicchdafj
fbhcacaagffdacjagcgjdjhccjabjjdchhgaifa 
#cgggadcfaaaifigbbdchigadiffgidiiijhedib
diddeafbcgjefbeacihccdheeiihghgcafhdhje 
#ciibfebcifghcahegfdhhegidfcagibbiffbaej
ejeefgbjjghdedfcicgjfdcidbjjegcahagdejh 
#ahddijjjdjbiiddgbcdabhbecggdgcgjjgafaid
gbjjaedcjdhdbbceeighfcjahfacjfcfhgcdhha 
#caccgahhbbdhcagdfaccaahbbjhijffhjbdiaje
ajeaggdcgcfgcagiechcgccbecgfbhbajeahieh 
#dadaejfeieidghaffebghiibbcichabjfbfcihj
ifhgeicjhdbajbdhiiibghfjdbhcbebifjaeajj 
#bedjhccbgjaijchchidacicgdfcefggieeiagai

bbbbfbafgcedibbgjjabdaiihhheijjhfejhjeg 
#jjgachcdjbdddeeehffiaaibjihiaafcaadccjc
hbgdcfacjcdicgchacjhcfdddbiiiiehcgbhcec 
#cficebjagjfdbebeffihighfefgbifaehfhdefd
jibcbjcjfefdddijcdecabgadbgfbfcafdhjgcf 
#jihbjaeaciigjjdfdfjhhbffefcehdbhibbeggb
fjegcchdaihhfiajiecjiiiehadidjiafgdeggj 
#iccichfhaecchhbhbiadjddffjiecaehdacfecc
fbbgcahadifbeffecdighcbfiahihaghbjhbiic 
#dahjcddcfefadijidjdbcjbjcedjbfcjcfjbbii
fafaajigijjbcccfacgjiiiidcgfcaihecibdbi 
#ceadjgifadddiheihcfaecehciabghcccechgjc
ghechefgfbhf                            
----------------------------------------
-------------------------------------   
,,gamma7#b_/c7 to #bef places           
#a4cedaagicidbfdjjdafidebhhjbhaedeacghee
aicbgbffjefgicfihcidjbbdfg              
#ifcghbifedjagdbedafgehcdadgiebigbiaaajf
dcdhbcfajjccjehhedadbbfae               
#ebehfbaahbffjgaiaadhaadcbbhccdcdaeeiaef
bjiagejefhbeibcffebccheba               
#aiajhehjaaejaggjaecfagjbchecidecfhcagge
deiigcfejdaeeicjfiachdbbh               
----------------------------------------

-------------------------------------   
gamma cubed4 to #ajbd digits4           
4#aibcaeeafhbaahdehiffcaibcgddaifceijgab
afghbfaccccgebchfgcbebiabeciaghhddiafac 
#giceicgcijigabcgheeffjeaafejidifeghjbdi
ggbiaijhdibbihafdcgcheiicdebacedcdbhieg 
#gdbbhhedfegfjcbdddbjffhhdegccdfadffihdi
hjiefhjhfhdjhgafddjcegdafjhcjhedbgbcjib 
#edchaffbgafggcajgichjbjjciiidbdbfiaeaff
jehggibgcecgiegacdjhghafeiigcgfgbhbfcje 
#fagjffjebaefgjibgcbdicbadjabijjfgbcajea
gbagffbhbiiihhhicfheaabdgbbibgdcehdeeij 
#caiidbfdebjaghechchibhiceijhejehgaghafb
jfcbegedajgbeacighahihcgeefaabcgiaejfcf 
#adhgfjiajajdcijbeijcddagdfijijicgjabiga
cbfcedfjbfhgffdacbhdcdgfijagdafcaggeggg 
#jhabcdfefdabcfcbaghfcdgcdfabfaegdgdcbhc
gaeiihjfjbbdhfcjhaichgcdiahfdfhdicdgade 
#jccbcgjgfcfahiebdgfcgdbjeieejgafddefgjb
fgeigeaeaggigdaggbfbiebiffidfddjgiibccg 
#abcjhfaaecfcfddddaibifbcichajadbfbgieih
eadghgagcdbhijiadjdfffifcciadfaaeadbhbc 
#cbedhjbefdgaefagiebajijdaieajheedheficf
egehdfggedcggjhfbhjihheaegdhjcahbfeidhc 
#ceabdfiifjcbjjebdefdbbecaggjbfbjdccbafa

eadcgijibeifigbeeecgfacdbgbfidggbbjfiih 
#ddjhdbbhcagefgdfbaacaeabfdehhgeccghchei
gageihbhibgefdebifcjacgaagcidfijhbcgbgc 
#igcfeeihhjh                            
----------------------------------------
-------------------------------------   
,,gamma7#c_/d7 to #bef places4          
#a4bbedafgjbdfeaggfdeabijihcjccfbhijebfh
eabcibdhajhjgjfaabcjaahich              
#bhihbbhhhdbfgihcegbcgagbcgfbadiaejffehb
agcchjbcgehhjccafcjaffeij               
#cbifajcidgicjdgajbeejeiihchbbgggiaibgfh
ijjggfeajafijadeeccafegia               
#eidhgeiddebggcjeaeicdbijjcgeghfchjifjdi
bchhcdegeihaahgcjgjaicegj               
----------------------------------------
-------------------------------------   
gamma99997exp7#a7 to #ajbd digits4      
4#bbdeagbeaihcbcbjfbffeabhbicgdciadbhfhj
iehagdfegcaehhgbiiigfddgdhijeibgehdfdgi 
#heabeafhaibhcfbegbgjhbjigcjchagbbheghcc
chjdbaagcicdegciijigjcbihebcabdcecdjiah 
#gbceafhfcfddebhabdcbbjajjcgjhchcjieccej
cbeaaidjffbfafiaeigihehjeechaiahijaifdd 
#ebecffhddjefaccebfchjejciidcfadabcadifj

badiiffbhbbjcgjggjifiefbegiejhcdbgidegb 
#gjbiiabfabdjcchigfdcgafdicfaabjgdefcjha
agaaijiaaabjbbbbgeachcfcjjdjgdgegdhhcdi 
#ddejgjhibjadbjcfcacfcejgfcfjhidciihjeai
ahgbfaceiaaacgdejjecdhjiifdcbeahacjbcjb 
#hggadhajeighegeggggjfgghdjgdajjdjjieggi
fjhjifagheifegcdigcdgfeafigjjgfehjjjaji 
#jgcjihdeefdfgadegccgfdebjfbdajiiidicbca
jhcggdaafjbjfjgjabebggaccfbbjdfdidebgfb 
#cggecbhicbfjaicihgjehebecddgcbbdhfhjcij
dccfjbihgijghafjajciibdeagfgfifedhhghge 
#bdihgiacjehdcjaijfciegjgabjihceejcbihii
ejfchfhefdifbdcicdaijbgabchjhhddgedebgg 
#ddhcgbjegghchgcbaabeddgcfjbcdhddbdgdjbh
jeceaaediagfdacgcjefedaiaheeefaffjcbfgb 
#iiefjfchfbhaghefggdbcadbeabiabaecfchefg
jehigfjgjbhhjidbdibibhjibjeacbfchedgcbc 
#ibbjgjdihgjbhbjbahgdcgfbafffffcgdjhfbcc
ihfgichgaibjhebgaghefdegebcgcihfiidddcj 
#djfbcjghiji                            
----------------------------------------
-------------------------------------   
#b9999sqrt7#b7 a transcendental number  
to #bjjj digits4  is                    
#a_/b                                   

7#b   7                                 
#b                                      
#b4ffeaddadbfijbbeahhfejbigbdihgcacihdhb
gdbaacacgadfeidibhceigieiccfdibjddfaghg 
#jeiedhfgfjiahjjjeaifdafidaihicfchedbcec
hgeadfgdbdbjcadchcfgdjghahfihejedhgegdh 
#iejhcaadghcifbhehcefahcfjhcdfabffdcagid
jiadhiajjecdjadcgciejcdbhgjhccaaijdebga 
#afigcgcaeiefebijefegfcbhdegbigihaggdcdf
cgbhdhccjhfbhaicdiebhediibgehcggcefcahh 
#hcjficchcbcddeifaahjejhjigfhgijhabfabgd
iajgbhigfgdbighdbffcgfcbejbcfifjafiefbd 
#hhagaafcigjbibfijcheiijceeefbhdfjaaefje
bcbjbddfejjffcahjfciaebiidgieibhjajbgde 
#ejjcebhdgdjhfbhfhefiggdhdiaggeadegddiif
ehhcgbigegdegbehiiijfchhbhjjjcejhjcfijc 
#gcchgijijdfgahbhjechcfeifgfgiebbhbidfha
hifjahjdicddfaecbhhjhdbcghiidgafhchbedj 
#efhficgaiigcggcfbcedcgbfcbfgacjhbjheffi
cejbdghacbdiehdddabffagjdcaahcchjiihbeh 
#aijcfcaaccghiegfhejiebcbgcjgigaaeghjfjg
fgggcgfgbeeiieggifbiigfidcichdigedachdf 
#bajfhcgadjcgiahidjegbjdfjgeeaedhbdhgdfc
aaehacbgjjeebgggdjehjaagdhhfiddjefebaig 
#cagijicejehacgjfgafbjeddegeiiiheificijf

idihgfighgfibdgbhggbcjcgijcciaaehiaeeff 
#dbjbfgfjfjigaacefhaebdijhecabdeeedefeii
ceegdagfabbhaeaeeafhjhiiccibhdadbafaggj 
#bcbjfbfbfhdbhifhgfbhaajhcedhighgjefgdgh
cecddgiidjhcccdafdbhjbbajjeiciffehadbfh 
#jiiejffefgcafiifhbjhegjjhegfabijddcgahf
fdechcfgejdfhifbcbehbgbacgjaabjejgdjejb 
#chfbebfcjchieccbfhcbefigcdgaihidcbcdeej
dggfdadaggdgdigdgcbebihgeeijgcjbfciehgf 
#jaghaahgifcddacfaecjcgfcejechegfbfeihae
aifgcdhdgegjgigciffcgfjhdfhgbjjdgcddaai 
#ehjggjiegidjfcdcbcfbgjecjggiegjicbifbfj
dghaiicbeeedadiegcehhgebfhgjhafbafbdbcf 
#ejfehiicjgdfggheefihgigegecddfejheeigbj
fiigejbaiihcjiajggfjhihjbfhhfbdhihjadeb 
#hiiiaabcffhdhgaagihgdbgiieddaafcgajehhj
fjbaajagadheghajgjfcbidiffdiffchcbaahfa 
#diedhehggbdjeijfjhjibejdgdhfeggfchcbhgg
becbiedcbjfhbhibefggffdiajahhcfehighiei 
#iighdigahhaifebfddfcaieaiajaeihgibcbhfa
hbjifagfcjhfgcaifhgcefehbihgfbjgibgdcga 
#dgagjcaeacjhgbegggaheiiebehadjcigegiicf
gehgfjghbdhfcbddhidchdibabfhihidgbbfecf 
#ddefchedibjidgbcfggghbcbicbeefhghfgfecb
ddifiaebbfigicgfhfjjcjjfhhhdhjadgfedjbh 

#dgjebfddfhcbfgbechcfeijfhadfgedcaiabhdd
ejgeehjhijifcbehcghaaahceiieibhdgfadhae 
#bgdabhdcbbhjddddbgidcgeabgjdddehiccceie
iaafajeebabh                            
----------------------------------------
-------------------------------------   
,si7,pi7 or the ,gibbs ,constant to     
#ajbd places4                           
#a4heaicgjeaihbdffagjcfajeccgjaegiiacfcc
dehjigbhihaaedijihjdghcghahgfihahijaffc 
#dhcehecbgajccfejbiedgeggjafhdcfafdhjjga
egjjicgbdejgiiijaifcicdbbgbcbbdadafejcf 
#cfejgdghhjbggeggfjdjgjjedbechgjdeidgjcg
edhjgjjabediabfaifjjjcbgjghegecabfjbdfb 
#ghabhjaeaeihfibgabfbeaeffehjcghaiagjfeg
jdihaiaaagadbaechcjagbhfhfijiejjbgffhia 
#ifihcghcefdhghficcgeibidcaiagecfahehhci
hgcbhacfaecgaaagdafjjeccfejbheihhibhijf 
#dadfgjjiedhhhggchbbdgaabieegcffgcdjffcf
eccbjfceciagfjdaceagbjciaabdjcjbhiaacea 
#deacahchfacdibibeggddahbdjgebfdgfjcjije
bgibjgghbadhefjbbahgahadijdbeddgaejadfc 
#fcehdbgggidgaaggdffacggefjehciihjhacfja
ehiggdjcegjjcdadjgeeiabjcgjbadaacihgjje 
#igdifddegfdbdcbgidegagbjbigiadfaieadehg

ejjeebabihcfhjjhcidjbbgeddjghgccgahijgg 
#fjjbcccgheiagdhaigcdfaeddaecedjacgeebjb
jfecdiecfcgjggdgigbcbbcecjgfbggaaajaced 
#fhjibfhdaagbdfbgadcjhbfgahgifjjiagdaegf
afhejhjdfddggefbideeifbghdfchahjidejegj 
#bjfcaeajhcdfjhfbifgfaaehchdbddfdbccacie
jbfeahefhhbddciiebhhejdjfhahjfgadahbfjj 
#ibfbehjhcbcgaecbbcbddfijjjdjiaibdbhighe
cdibchcifdafagdiceieegbgbdjdidifhbfieeb 
#idfhdehjijh                            
----------------------------------------
-------------------------------------   
,the ,gauss-,kuzmin-,wirsing constant4  
#j4cjcffcjjbhihgcbfeheigddhabaija       
----------------------------------------
-------------------------------------   
,the golden ratio3 7#asqrt7#e77_/b to   
#bjjjj places4                          
#a4fahjccihhgdihidhdhbjdehfhcdcfefchaagg
bjcjiagihjegfbhfbaceddhfbbgjebfjdfbhahij
                                        
#bddigjgbjgbjdahiciaacgdhdgedjhhjgechfhi
agebabffcchfbbbcecficagicahjjfjgffgbfce 
#ddccchijhfeieiciehbijefchcbbffacaiibhbi
jbfghhjfgebjhgffhibejagaafifbjgjcbbbajd 

#cbafbfiedhfbfbifcacfaddchadigehgjabbjcd
jhjehhgieddedgdibdfahefiecfdhfdddibdajd 
#dcbjggacddidgjdiefehdfghhejihgdcciddbba
beddhggjffdghjiaehhdfjgdiihhgabdjjgfeba 
#gjegeagighhcdaffbefbdidjgehijfigjdjjjbh
abajdbgfbaggaaaggghjecaecagadajaagjdfff 
#eiiadffigihgcagfacefjjfgjhgdhjgajacagie
bcfhidbgebaidhdcecjefghcjjbbhghefiighbi 
#gghcdghdehghbbhiaajigfbejjcjbfifaefagjj
bejdfdcchbdcggfdhfajbhchcabfhccjcgbdbib 
#fgebfcaafeccibdgcafgaaabaaehhahfcheacca
fbjchdjjebbbafegiabhffgebidfedijfhaacag 
#aeiicdcbceigcdidihejijdjidgfbacbbbihaja
gbfajgjeifaafdefbiijihafbijeeebjhebdgij 
#cebdjfjbjagbgiigdgagecdbgggeibgghfbefai
dcbjhbgejeacabahaefbheeabbbdhjicidgabcd 
#adeagjbbcgcehjeggbghfafjjhfhhchbiebcjde
ibfdghghjaghhiibaiijbgjggfijchiecbaifha 
#ihfaeadcghjcadiigdaajfibfjhhfgdbifbbfge
gefjebcagbgggebjcecfacicfbajgfgchicgfde 
#efjfjfjeibafehidffgeieeaijjdjjeeeijhiej
bbiecjidbcabdhbceebabbabdaedddjjfdgjcdj 
#efegcdgigffcigbcididiidfehdeghhgcjcifbc
jijcgejcciichefbajbdbcfijbeachfhjdadegg 
#iiefihabbddegdgaghjcdagcabfdecbbjdafcig

bcbacdjddddidhgcjbcaedagfgfhicgebajcjfh 
#gcghhjcddagjjicieddjifbgieehihfghgbcbji
eabdbfhiceegcjigjdejieiefhddjageeeaihha 
#ibahjbjfdjebijeeahicdidgeibfjjgcdhebbhb
ajajhhaidfddeddbbbcahhiacaibidfhifbbjjb 
#cjaddcggjbfiibcjjghjcjhebfaahjgedeaibhh
ggjejbajifhdbdicfbgaceibeahgfjggghhdffe 
#hcfaejbchiacdiccccabbcajeccibcbacfbdcai
bfcgbhiajfgjejcciibhbbfebfceefbjijbigih 
#fdbdgbgeiggbefeejhfaedhgedcegdhbfdgahad
adeabgjjjfjbchijafbjgggcbbddiidcecjhhii 
#ijiejafhjcbhaabaidcbjdhaifdchgfgehfccad
gihegaiaacighaecighjgdgfaejggbbaagejhbf 
#idehfcicbjdefebjihifiheeefghadajfifhcgb
hhdjehgdfajccghajedddcijidcfhcehcehacha 
#acaafhiicheeegfigedhdadiaddecdaejiabied
jgjjejaidggedhfafcjgedbbfdagbicidfhjcfg 
#caihjehfahcciahcbheiiacjcifjgbjaddeeiej
ddiggibabjgfabdghefdeiafafjhcgjeidihghf 
#jjfigjahidjihhfdjjgfddcfagjiccdagbgjiai
adccfejacgaegffjaadhjchadcjfbfbchjeadcb 
#aagcdhaeajjeeijacdefajaahjjgijejfchadba
ebgjicjhehhjibhgegjcdejejghjhadedehhaii 
#jfccfabihbgihadaagdeccibgcabjhjibhigbgi
bbbacbihjfdbidfhghbdbgdhgdjagdejeedjfgg 

#hgegjhcbcgcajigeiaeaaggfbighddcbhdgdgij
hagfeahjigghgbfhdafaagfcbejchfabaabiadc 
#fhcdcgfgjbcejcgaaafccjgbehfihhcbehgajcc
fcbbbchajihjijabaajaihiiagfhdadiageabcc 
#acdjaebgcchdchcgbcdejjicdghfjdigibideii
aehbbjabehajdeihbcjibeebhgbabdacgjdcfad 
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ifbejfgjbgigebdcajaaicggcceedhecgjaafid 
#fgdhehhhhcfcjhjcccbhggciegafefbgecdjcfg
bgbahjgjefbbefbcbfcgdadhhccdiibhiigjbeh 
#iggbiibbdjcfidagejgdcdbgcadaidaegdcbdff
gideghehfjcihidjgejigcefcfcfhhhaefgbaei 
#fgfcedchjffeeicichicdchbjgeihdjfabafjfd
cagffddbaijbfggggcgiiadeegiidejcadfhgjh 

#gafbffbbfebdacceijefiibhdidjjfcgbgddijh
hbafcebdbidhjbbeffccjdeheecfcfccgbeagfb 
#jdijgdfbdjfbichifbcijfbbjcjdbdhgbfhhdcb
cggfcagccegdbjegeciigegdcgcejhdjifeggib 
#ahjhhjjhidbjeijffbegbghbcjgfibghhfefdde
efcgehjabffgbhjiebebgcgihbhjcjjgffcfigf 
#ibhafdhddfeabggdgchbbcigjfagchefgejgadf
fibgdhbbjcgdhhaabbefciidjgebbgfbfdfdiid 
#fehdfcfgdjaieeiigcgjbhchcicaaihhdhbbcce
eciifdighcccafejjhdfgdiabedebbiefeabdji 
#cijifcghdjiedafijabcdfgecgebhjacijhjhcj
hfcjbbfeccebchgjfibgcjgaihdfedfdidedigi 
#ajaacdbhgaedfcffieedcdcgdfbaedciahhfebf
jhecffigdcffecjehhefbafddaafdhjfhiabhcg 
#ceggidcdaecjfjidghdegbgjihgjcgigfibacdf
bjeifiechhdchbfgfjhbgfeiahaggcfbgffiiah 
#gbghjcgedbaiiedagbdbhccegiajfdebjfacgcf
hhdgjhedeafehbbaicaehfdecggjahcacdjahah 
#hbgbeajiiibbiagfadgaahfjebiagfeeadbbhha
abceffbagbdafibfhjfbjfdhhdecagfaeafdbgb 
#iecehegihcgedabcgehgfajjdaedgehjeeiegcj
abbdeibgfgaahiebggccchbccefjdccgdbjacba 
#cibhjdcagjeccgidfcfdfdbhceaiicjadegfgjf
diahdggjggfhieihhedbafdgigccgagfifbeidc 
#ichfdhjgdhicfccbjajihhicfdcebhcbddidacb

eficagdchcbcejibehbhfdbabgfbjidgcdcbhgi 
#ihdchgaihbiafbejcehhfcfhhegddjhifjiafai
gfgeecjbcfcfadghdjahfbgahbggjhhiacfjcih 
#iccjggbicjfjbifgaggfjbehdahjcjaccdgedgd
djfjicbahbbbffbjggjeihdbdgfjhbfcgidachh 
#eihfjaicebjhieihbaidahhegbchbcgadbgaicj
cdicedeahbdjaabfgajdfjgcjigdabfhabgijgb 
#gbfdchfhefhaeddgbiaddhbfgfachiidejibjfd
jihgibfdgfibegdfihhabccdfdbiiebfgcjhbcg 
#djegbjdjfadcgdhgjjhfgjdhfabeiieijaghdbd
igfhdehddgcfhbdhbgidghbdgecagfcchagdhad 
#giiegajcabjccifcdebbfgdcdaeabcgbbcbbded
fbfedfcbhcecefdbdffbgghfdfjhcihgbagiabg 
#hdcjhifdafcfdbbbcgaebhbbaiihfjhejfjjaeh
bdeafidghcahibfjfjafehbgdiaadbggdiccejb 
#hfeejcgbgfiajfhajgeeghbfdfccdjciibaibbb
fjbbjheijifghdahfjjacheifechggbfehbfbdd 
#fegeigfidjfibdjedahjddddgchdgafjgijaddi
gdcjahjeehhiccgfbcgfabifiahbbibcdgfhdec 
#geieefdfhdbaabbfihgcafcgejfbdiigaahbbia
dhefhifjddgbebggfjjicicdcdceehcciaieafe 
#acbihefbcfdehicadiajahfjhdifhcdhjcchjij
icbgcfbfajfbjedgieigjdbabihffihhcegcefj 
#djdcdgabhciihjabdihjbbjidffheajicdijdjg
hghdejajbaagfhdbgfcdejgiacgfhgfjigdfijj 

#ffegeifhcjdceaibfffgfefcifjibbfdhhdefgj
babhejgddhbaahdhcfajbijgfhiaifdicdjbcjj 
#fdagecagcdhciadgehiaffgbjbcjfibdecdgajg
fbggaigibebdiigcbhegfhijchhfhjadaghjcac 
#giidhcfeajhiebgbbjidfeiacjdejffeffehbeh
eciagdfijdhfhgbfdiijbedfgfeiffeiiafdedg 
#cfeacdbeigeeeggcigcdhejfebhdciiggchddij
eacijehbidcjacjjjhcffifadeeffigdhecggic 
#djghhabggbaegiadhgbajgaibehhfijhibgghgh
gcbihbihbbadegdbccbgcbfeihgihbgefiejhih 
#hdecjfbdjbbcjcfdhfcdggbbifgjefebdabgjce
hhghcjbhaidjjgdihjegedcijafbheghfgdeeca 
#cbgaigfebfjgajgfdcaecaabciaebfjggbaicfb
addcdfjifjhigehgbficdbbcfgdccafacgahegd 
#eggfjhaaggeaeahjfiffbajdgieeheadjacjjfi
gjahdejjgjbfbijdgidibegjhcgabjagebgicgh 
#eediegfbgciabdeehgadhccbjajagjcfahdjeba
fcfhahjagcdadbejhihjfafjfcdfgfccjhejejd 
#ahdehehaffbiccdjicdgiaiiajcfheiacjecghi
dhbaehfeagjaahabajaacccjjjffieggebcbghf 
#fheeahjghbefgebhcfadididibjgdehcgccfhde
hacfiadjgiggeieibebfgbgciffdbcdghgdffad 
#ciihaifdhjhajcfgjejffjjebchbfiafejeeadd
fcdgaaaafhfgdbhaggcaiejbefjfdbieafcgiei 
#feidgefddihghiadfaddfibeicffbicjicfdhjd

hafagdjeihjhbadbedcdjebebaacgaccbdjhaac 
#iacegiigafbbhehajadaiajcdajdfjefibijghb
dihiefbadefjjdajdefibbbadafhcjhicbcfffb 
#eagfahfifbgagaidechediiheeadhdbgeagccfi
bdabjbfhjaeiibhjhcbjadehcjjgeddhdgdbcca 
#bfdchghjhdghjhejefajdcjdijiiiicfdcdeije
aieahgdidhdcfifggbgegdgcceifgjhhccdifji 
#aegddgdcegejcihfjbjafcigfffaadbgfecfieb
fgjddaaeebjjaiciadhdbicdfjajaeabiecaagd 
#dehhgfdhcjgjcgafggcifaedbfeeiaciijhcjcg
eggffcjbacjiijhgabgaihhgjfijcbicjdgjabd 
#ajehfaejfciihebiihadageghjdcjcdhjhjcehh
bjcbjbjaajdgfjgjjdgeegajafidbcdabjcdajh 
#iaefdcidghbecjcafdeicgcjdcgeehaidfhfgeb
ecdiecbcjacjbgfghbcecefjaaffdacaaaggiif 
#jiigicffbjdcddiefifhcedgicjgedcaacbgeeh
fdcahigcaeaeagajfddcbahibdigicbgghjabfd 
#ifdgfddgedfgjghafehjgdjfacabeicgebgahdg
djhhafaaedgihahcjghafgeajdghjibiadacied 
#efdfcaafjehabfijeagecieceefiaeggeehjdaj
fgaihabcafchdjebggeefjebbgbbbcgfdgaahhc 
#bccbbcjiiehejfhigajahgagejdghaijfeccdid
hehdbcbeigfbbefegehdahihebiaddgaghcceag 
#cbbfjbiheghfbibidcdfejefcfficbafbfbgfgc
hafbdeiegdagicbfihhibcbgbbjffffcfjhaiib 

#dijihhhcadfhebiidjiiachfgcdddfjdifgjhdb
ddbighbdcfcjbcbichiajceeifefjagciidbbja 
#ihhfijbegbdedgadjafccjjifabadfahgbjhcfe
ajhfhhaheccdjfjfbbjagjiieaehbgjgjddbccg 
#jdbahjagffifcdiaccfieiifjfdcbbjjecbhhgc
didhicaceiffjcjdbdchjhjdefeiddgdcccefgh 
#cafgbgjcgbifcfcfgeidbafiiicgiebb       
----------------------------------------
-------------------------------------   
,the ,golomb constant4                  
#j4fbdcbiihhedceejhgjiibicfchcajjhcgbdda
gifdbfbjahjebibhf                       
----------------------------------------
-------------------------------------   
#a4ghbbacighaiacfiaaaggddacdebigbedicdjg
iagcaijiggcbcichajbdieiief              
#hheaedabhgfcghdjhjbdcafgffceghbeecjhicd
dfiafedcijeijbdbhcbjjgbcg               
#afjidffbdabbdfihahcaiccjdddjfiaehjhddcj
jhcafbcebadhejeibbeabiaah               
#ejcbbjegegjiabfjfbjebhbffdfbdejhjbgfhfd
ecghdbfafhggaidceghchcjcd               
#eighidcfjdfedagicccjehafgjcjbeejdgighad
ggdjfccihddbiceeeabeigcib               
#gacaheadhdgaeaiidfdbfgjiceeegejhbdadfgh

dhcbadibfgdaighcajegafgbb               
#hgfhdfhigbffhcbebiidhaabjejgcibbhdafcid
bfgchgheigjjjfiggiaajdbhi               
#hdhghjcafbiiifiibficjdceaaigcbdiihgajag
hgffgbgagcjdehfcfdheadgfj               
#dgjafeicafhhhaadicjgbjfejffjgaadjeccfaf
jjbffcagfjagjfhjhedechbfe               
#iaejcahaijijcjfjaiceaehjahdbjeaehdefgfb
ffdjcgbjhiefdfeihgifdcegg               
#bdhhchicbedcdfgjiaehacfehjhihbiahjaiijc
gbhfghbgigfagjdfgehagbccg               
#eejgcdjgegbgcidcjjgjjggifabeeahbajhighd
affaehhecjeidcdifidecagha               
#fhfdehdgcdffdggajchcfgidbbbhjheecaacdea
fifhggbjacfeijheaeieediga               
#jjccgjibhfhcecabbibbeebhfejbiebgdccefee
dffidhaijibaiggfdgfbjfgag               
#fhaghiaegeedddhijaaijhdiaeadccabjeajiij
aceadgjdecddadecdcbghgfda               
#fcfgcaibcfcagcdhdgeghggecfbfgaadjheeabh
cibcbdjdjchbdiejijejejgii               
,grothendieck's majorant4               
,pi_/7#b99log7#asqrt7#b7772             
----------------------------------------
-------------------------------------   

#a_/,w7#a71 the inverse of the omega    
number 3 ,w7#a74                        
#a4gfcbbbhcdceahifgajbbebjaggfieagjgjhjd
cfjagihfffgdgcfcdegjdefije              
#edgbgehdgahfiiegcfghijhchiajejfhaajeefa
iccjjbjbgdjedfhjdfgcgfdjj               
#bdjacgiebjegchjajdccibfgccjbcjgjcfdigeb
ifgeddgafdbgdcgdcjcijagda               
#fefechdjjeebbjiebdcdeceeffihbffidbfciee
hcjbjecedhhedadeiacejhbjh               
#cajdcicfdcfefgcffahehgjdcccaegcagahjibh
gjigghihajiedafhcfcgeabaa               
#eggdgaiajehgfdhcabhcaacgadcciddebbfhdha
cjabjahbjihjdjcgiddjdbefh               
#geehiachagdgjghadaehbgdajfgfagfiigahjaj
faagfehaaecddhgaabbdijdjc               
#hcidhaidheaageaahbihdcabcgibedjaibeccdh
gddbfahdiieeccebjbiigghif               
#giabhhjeaaifedegfieahaaigghfidgibjhdcbh
bbigfbciidfaihhbjidhddeha               
#djiifhjfbgeejcibdbijjddhichghcfahajgfhg
aejgghhcaidfgbffhjiifeeci               
#bbggaabebhbeeiccdcjifiehafigghibeeahifd
hdcbgjjghgaedjjahfebiffgb               
#hcihgccijdhegiefjdajdccccbefaiahibddfai

iacahicadfejaiibgfidibgig               
#jdaehehhecbcgfgcebeagahjijjfjigdbgdddbh
ajedifegddhcgdhajffeaahii               
#ejiefghbfedgeefcceibbbbeedjcbdcdjhhbaai
ghhefebgcjeahibdibeifagdj               
#dcgegdjhigahjbcifihaifabjghgbahdicaaifb
ibaficgbigcfeffgfjbdbgfdf               
#hcbcfdcacgjbdahajgggggheabdfhejdjjcaaca
ijbcaaahcbcaeghggcieibggi               
----------------------------------------
-------------------------------------   
,khinchin constant to #ajbd digits4     
#b4fhedebjjajfecjfddecjigadhcedhagiefich
bjchbbiciiddfbiecjeaaebcde              
#eegbahheiecgaebjjbhjaadaagdicahdgfigiie
aecdfeijebhhjijjhbhigfggg               
#afdajifcjeagibeccdhcbeiffhchahebcaedbac
cbaaidiifbfjcicbhebbjddha               
#idjifahjfhffdaffdbhicjhdgghhjfbjcfjgcgj
ecejajccfgbfcceggbhijdiij               
#dbgjgjbgbcdeagjbfbebcgjbcedehajfhfcahej
ajcbcgdfeehjcggejbfddbebd               
#hebhfidfhbcdahiidiaegcjffahihgbjgiidacg
bceejjjegicegcffihicciejh               
#gijbabddfdbjgebhigdadeiadgficjahddijejf

jagicdiichebbedgjdjdbjccg               
#gihefcihcajaegjijbbbcciajjjjbbjggbejife
accbdfjddddciaiafiadfjhei               
#fhbcdhbabhcbdfbbhbibgajabfijfigdahbcdhd
ggfgedegcdhihfbedbjccibff               
#bceahfbjhfgghacffejififehcadfiiebgahcgd
dhjedjabaiecfffffjdifdhbf               
#ihijhbgedhaaebedgbaaggccjcaifgeidgchcga
iciceghajfjeibcjdjahijgaa               
#cdifbdfgcgjfhdabbagidfhajgdjfjhiahbgffi
effgaagaffhcgdjeijdgcicfh               
#hjiecdejdhiiigjdgagfcijdeacdcbcbcggaeaj
cbaifeaejchbdfihhhhcbdhgj               
#iceciidfifjhbfdghahabjeffcdidfgabeghdcf
ffdegigdjigghdhcffbjdiggg               
#gdhfhbgfefigjhgafcaibicheabhiicadaiieah
faafgcgibfedfbjefcejeieac               
#hegacgfafigabfhgbbiihjecbgfgcbghgajeacg
fciefcgaijbcadebhijjcjeha               
----------------------------------------
-------------------------------------   
,landau-,ramanujan constant calculated  
by ,philippe ,flajolet ,,inria ,paris   
and ,paul ,zimmermann                   
4#gfdbbcfecehibbjffbiijfihgcabejjibcbhaa

fgijedacicdjieadgbafhffgcgdifadfdafehgc 
#bhehhchdjaejejacacabccgbaicgbfiabjgibei
bfcdahgdbjfdfghjhdcbcjfccaedcdfbichjeca 
#fjeagaafifcfaggejhhaiifabdchbdiidbggfhc
dfijeafbceacibahgaifbjefijecbiefddfgjda 
#iagfcdiggjfeiefiijegabichffjbhichehiihb
ifajeaffbifjhijiiaggibihcfjgbigcfigbjjf 
#djcafiheabhfcfeagcdgcibajfegfheejighfha
ihafgdgjgceijffibahcjbhhgeaejafhifbdfdf 
#gajiahjhagajfahjijjhfeagdicgiijhbdbjdej
egjfffbjdhihfabgeggacccchiedhdcbejhcjce 
#fhbiejdjggbaeigebdabadcjidbdgjiecaaegfe
eeidjdjfdbbiabeggbgbdjgaefcdiabahgbcbgb 
#eeefdjhhiiiieabgjeacehdigbheebcdgfdeidb
dahejeiiifcehjjicdgcbffidaaedhjgfiaafga 
#deehacjbhjffhiheicafgdicfbfbiebeiefjafc
baehdchibdfchhgeehcdgaiciichfdehafihgea 
#jdehiceahgggidehgbgeebbfddhgjiidcejeeie
idcfgabiiiggghjffihhjefdeeeibacjjfijheb 
#bdbhfgfiaajbbfdebgecadeehafjhhaafbifiig
jbihgficgjidchhdbbjhidiebijgiafbfcfcebg 
#giadcbbhfaefhfcbhdbaeiddhiicdgahcgdhcbb
ijdaeehfchadieabhaebgajbjfhbdibahfdehbg 
#ighadejihhgjcgibaahjifbihdjidcfjdhiabcc
ibdjadhebeadcbgdjgibcffjaghecbgjgjghhaa 

#ehdiddjdejibecieaigahaegjheghjijgfijggb
aibifbeebbfbhijebiifgbjjeajffifchehdbjg 
#feejhaffjebgacbeeagfaaejjicfaijajahbaeb
jciedafbagdddgdcefegacadjbfdifjeadhjcbb 
#dciacddegebhjjigcifjdifgaijgcdffgcihfba
abgjcdggjfbcjidghfdfcgbagggbdeeeaaiafji 
#ficcdiehjaafejaechadfhiggcbidgdjjbedbgb
fiieahcgchhabidjjdcijdfejejcajjiabajcfa 
#ihjecegfjiebbbhhcehdgffigdcbfgejggegcgi
hdhiciceffdedjfjagbeaifbeachbffgahfchbh 
#hbbfbifegciidchfbfdecjghiaceadeeeaacbjf
dgeidgiacbdeehbdbcffbdjeabfjgjchbefjija 
#ihdfadegeaebieaeaaidcbaacefhaddafgafjjh
igdchdciahdgdjbeijhbfdiejacfjbhcdjjgbfj 
#fcdajhfeigifchbeifghdacfcgccggfhjheghca
bgiadgajfdagcgjegcccgjdjadfjbdgcgfdhbjj 
#gfhbcaaahdijeehfghiidajfiiegdcibbdegjhi
fffiajdiaecdjhiejjacidaihijifegheheccfh 
#ecaiheffdjdbcejdidgdfcbihjddhaeicegchch
fhgdadbgfiaefaaacdgghfabbijhicigfdcbacd 
#bgihgibjfdgbchadicbijedfbfdhbdijggffjcj
hhacdhgjeccagbcccfdjgbihiidbdefeffaadbd 
#jcfhbdhabhgcieigijiaegiighajfbgbcddfdbf
cegbccbcdabghcdghjhcfjbbdbdbabijabjcaii 
#fihdheieadbibafhghhdjgaefbfhhgjcdeagdcf

hiefciaagjgbfegicedjgjejgidadaajjcdccie 
#ehbgifdjighchiaehcjbjdjgedhahifbcbdhbhj
dghbiedfebcibbchgbaidcccihaheabeajjdgdg 
#iaefehbdgghbjiffdedbhacbdjefbjejdhdafbi
fcbfhiaegffdadieidiegdfcejefhidhfehgbhi 
#bdcdacgciajjfjhciccdgajhdeebicfefihbice
cchebahadcehgdfiibicdhfccacefehbjcjgaib 
#jebifaffeggegegbfffbgdjhdeehcgabgidfgii
ahjijdgajbeidebeaiifhjafcgbfcabfgjchjbc 
#ddgbihcjieaefhhfhdajadcjhdieidajhgighjg
jacefaebdjdihdgijigadcfbccajeiefiagidca 
#dcafiadjbaaajdiadbiheifcjeceafggafjjjhd
hfgbfjhiecahegebicbhbahcgedeehiedfffddf 
#aiadfhbebcadhgdgddiifdjajgddjbffdfhfjji
egbddhfgafhgehbfjecaagjfefcddhdidhjjhda 
#jacbcefafcdebihceeffahhccigicfafcddjihc
cbidaeceaffbdgcffhfifehijagejiibgeajdbf 
#agidjbdfebbhhcdhhcjgabgdgcfbehffeabgfej
fajedfjbhifdiaabhgbhghfbgchbfceifabiccc 
#fdfecbaiadfbaicfddcgecgeihfjebbbdajccdh
dbbacedfacchabhabfggbacabdehijajajgccga 
#hccejgbbhdjaebifdaagdagiifcijcdacffddbc
iaijbghiebehdhfhhbieifhifiedadgccggaadj 
#hhffjhgceifbggehgfhjacebdjjjbibagibcebh
eebijcjfdejhjdbbfigiadehiihajaecbadjheb 

#ijfebbheaaeeibfefbbfhhdcjigchbbigbiafga
fdcihjeadiedhfhcggbigaagbfcgggefehhiaif 
#hgaafjejijbabaacgafeaijbcdajhifeibicgcj
bdidfeggaagejjdbhbhjfcgbedbgaahjefajjhf 
#ehbaiicecfgabdbhdihcheafacbhfdiijdbggaa
bhdjgafihghghccjjfcigbhdibbhbjcddcdidee 
#cdgfjjbahjjcgeicegajcbjejfddjfgefgcahag
hccgagjcebghcaehbdjdaahgaedfhfehaagahfd 
#iebgjhiifbbdcdhdiggichhdhcbhaidighfcefi
afcdcgcccagdbjgbajjagjedcjahffiifccfedf 
#cdeceffhhhjcbdhebhjhcgeddibedeaggicgiji
dhjgaiegabibecgahdaaafcdacgcjcgdeceaaeh 
#hjhecfjehchgidgbddjjfjgfaggchababajbghi
bcdaiccjabigfedffgedhejeiifffcgggfdcdih 
#cgbhbacdjhihjibijbhhibghjhhdbhceffegiif
cebhdiggfadaibeebigbdegagbdhfgehgacfdah 
#bjacihbddhcibagbgbcbgfdccihfejgjibdejjh
hbjccgdiidaedgjdjbbddhfjeafibjdiedjchjh 
#ibgihigaajeagaggfhffgcjddjhedeahffffbbe
bceaibaahjbafcehibiaiafghejigfhjadbffdi 
#cebdfebdacaebgiceagbjhdgdjecggcegeghjdb
ciihdjhjgbciehaeaghgjaibiaddcfeicdjcdbf 
#jdadgbcfeideccbjbhfhdbjddfbjhgjagieifgh
ejgajejddccehiihhebiiihjefdfjfhdfghgbbe 
#gcbbadiegihedigcjibdffbbadefjgceecjecfi

abbagahbbihjgceccjgigjchggebibdfjaghidg 
#jbfaaaebbcfjgjiejadcbffghhhaacbhaiijhcg
eacbfhhdeeeaeajhibaijgiajjbfajheiiajjbh 
#hdahifecaijfhfeeaheigcgfhjjediaeebgibci
jaefjajchaghjjeecijgbhahifhabfdhjjbbgdj 
#cjfehajaddeihicfcejeidifbdaebiaahddgbdc
gcaddhifdgijhgfagjcbfjcfecibijaccdcjfhd 
#ghahahbjfiffhcgfjagcbijcjaffjbgfbdihdbb
hbgeghhdaedhdefcadgacgddihgcbeegcdjehjh 
#jeifhhafhghgedbajifachdhdaeidgeejbigehc
ifhcajiiehjgcghgbcegggghgdbafigjdcdadbc 
#higdbddfhdebhfhadgcejiicbifaddfhdedcfhh
bfdifhhfiaibafafdifgjceefejcffhdgbfabcf 
#cddcagjfjiaajbhifdejghacaafcbgbdijaagic
ffdfajgbbigihbjegdjgeggiadehjjijhcjajfj 
#iacbafbjjhigbdicgcbjajcbggggddgheabhgfd
chhjhiihafeficbgaejdjfabjijbbahbgciceih 
#hbcahgjdfchcabbhifibabafajdhdfedfidhcai
gibgijcbjeaejgfgbfgcbaaiafcbahbhdcgaagb 
#jcfjdfgaccjaggbdjaecdbafeficfheghhhgghb
aaeiihihagigedaedfbhfjjjbcjgafjjjaeiihb 
#bdhbggbabfbhchdhhcjdgidcjffcbhedfhcfbbc
iddcaaiggfeehdbgidiahiicebbebaefcbcgeee 
#fejadebhfbaajeicfehfhagaciiihdjfcgajaif
djdaeidfbfdhbceahcghbafbcefehiaffchgffd 

#eiijdidehfbgibgcggfbjdhfafjafadiejjdbbh
ighjihfdfdjgjdgafbjjgfdeddefaicabgchgdb 
#dgjjjjdgd                              
----------------------------------------
-------------------------------------   
,the ,lehmer constant to #ajjj digits4  
4#eibfcbgahbjafcfaigajdjghfjdiiegjadfijh
dbgedjgaigafajgajiiefbfjhaehbdgcebcfdaf 
#jjjheajffdghdbigajabegjeaahgahcdfedbchf
ifcdibfjbigbjfgfjfhbghefjgihgaigidcgdhg 
#beadjcdjcjjjehcfibdhfiaeechfdjfadffjcdh
effhbcdabaedhdefcfddffdbhcjicghjdihjjfh 
#iiaahcahdifefifagbjabaacgeidgjfbcdcfihf
eadeggaebjbhfggcdbagbbiehjeafhbcddddggf 
#hhiddejhjedafcbjfaeeheabcbcfbcjfdcgiagj
bbcebjghafjcffifcibgfcdbgabebdhhedefgdb 
#fbhcigbcdaidhfaadaeeeddaaahfeffjgcdgibc
hggihjdcdffcdbeecafibgecibbiigcfdhfcdhe 
#hgabahcccgfgabeccbgfeeebigfebhfffabfjff
aheghhgacfbbididjcjhijcaihdcacijgiehdje 
#dghihiejiidgjjgdhibejjecfefcbifijjagffd
djhefceihchebjidghcjahbahfhiebacghdhihd 
#ifggjeghgcefecgchedigeicebjfgbbihfjcicf
bjgebhegfaggeajighgiefhffbiceaiabghddec 
#iccefhdedhgdfadffgiicfhhfgahcegdcicgcgi

afeijhdcgfafegjdaegidgejggfbjjcjiefjaae 
#gdjdbgfjcjbdiciicibadifdaicfgdagagbjfed
bifiecijcageebfieaecehdcdciefgbgaiaeaij 
#afhhhjihagageigbdijfgehahgceeiiijechedb
fijegehiefijafbcjgagdhdgigbgjahfjijhdif 
#iiifhiegfgeifjhbhdihchaifibgjdjdaiejcfe
aacjihjjigbfdccjieiihbcffe              
----------------------------------------
-------------------------------------   
,lemniscate constant or ,gauss constant4
  also known under this formula4        
#a_/b99,pi7#c_/b7_/;,,gamma7#c_/d7b99b7a
_/b72  also known under the same number 
divided by sqrt7#b7-- #a4hed444 see     
,d4,h4 ,lehmer ,the ,lemniscate         
,constant 3 ,,mtac 7or now ,,moc7 vol   
#c4 pp #eej-eea 7#aidh-di74 or          
,abramowitz and ,stegun1 ,handbook of   
,mathematical ,functions1 p #feh formula
 #ah4ad4g ,dover ,publication1 ,new     
,york #aifd4                            
#b4fbbjegeedbibaaihajdfdhciehihiaaaidacf
hbgedieadcafbcafbhafhbagjchjjgijehgjgjd 
#adbejbcjbieecbifadbijicddfacegebfgaghcb
ahjeefjhiefijacicicefidgjaaaidcdggebceh 

#djdbbfdadigafdijfieaicfhiiigiicbadfjgbc
hcabacijhajbjfbbahigdbifjjhefeedeciggbc 
#jecfiedigajbhhhhhcbeebe                
----------------------------------------
-------------------------------------   
,the ,lengyel constant4                 
#a4jihfhehjeebeahgja                    
----------------------------------------
-------------------------------------   
,the ,levy constant4                    
#c4bgehbbiahgbahaaaeighgfhahhbdechdchfcf
jhdgeebeihbcgdadidjeaihibd              
#aijgbcbaefddifjceeahabggedjdgiagdebidib
fihebfbdcdjafcccbhahihjhe               
#aaejcdagjiigjhbcjdffdfefdfgjcgjhjgabijb
bdahfacieidbcggbjabihagib               
#dbeajhgfigfadicjjbhhjfhbdibfagjeidjdabi
jhjhfigjedddabcdibbcgihhh               
#ddbgjhigbfiafdjihceecehedhjdhcgdghehbhg
fbebfiahddejjgfdcchcgjchg               
#fgdahhddeidbjceagjjjjcgcbabbbcjfbdaicfj
acdjiafegdhjdceddgjbbagcb               
#eiichbbbihchbbfcbcciaeaeebheggihfabjdfh
diddgbebadhgbbijadhjicfab               
#jhijdedcdbjibjiieaiighafbdjjeaejcijeach

aechggadgedbiacicjgcidfag               
#hjdeebghchbgggbgdadhdgbbaeacdihhbffgbia
abaeddhbgjdebjahfccdhgfej               
#jaigdfahjdahbfbifecfebhhjaafhaeaajegeac
ejhgaaebcjafcjbidbgdhhccf               
#dfbfjiicfbhiffjfccfcggfcagbafejhhcdhjai
cfaaagefcbffihdaaadfgegaf               
#dfgcghhecffjijfbdcfgjccbeifhhgcgeceghad
bfedaieiibejgeebghbjedefc               
#gaeidjgjaggfafcbaaecegjeeibccibcgidchdb
ibgjijaggdcibcjafcgddjfjd               
#abfghdifafjgdghcjiijbiifhjefjhccfijdgij
ibcbccfaaajcdibgadgcgijje               
#bechiediafegadgcdaiigbbicafgffdfcjgfdaa
ihcbgfdfechhafifbjdfcdfbi               
#gccbiidbhfahacehebjjejcggaedbffhibdbdjj
egcaeaheegcjhiebhgaghabid               
----------------------------------------
-------------------------------------   
log7#aj7 the natural logarithm of #aj to
 #bjjj digits4                          
#b4cjbehejibiidjdefhdjagiiadedfhdcfdbjgf
jaajadhhfbhggbigfjcccbgijjifgegbfjifggc 
#ebdhjbceiigbjejhieihbihcdaifgghdjdbbhfb
dhfccdjiebedfejhbhjfgeffffbhgcfijihghaf 

#hidhbijgbjhcbeeedfhjhdcgiihidhbfbccaihe
bhcicejecjhifecgggcbfbhhdfafccffbbbbhgf 
#ihbaihhfgdfedcffgdgddjdbdcbgdcfeaeejdhi
cdcadiciciadgifaidjddjjbbbajeajagadagdh 
#jjcfhhjhdjabfdgjhjfheefggdcbafbbhceebbj
aadhjdffcgaefeiabacgcdejgdghefidgfhcdfc 
#fafgibajahjfddejgjfdhjjjbggejbfhdiafgdf
eejehfheficefgcdbjfgjehaacfdbibbdeeddje 
#gehibegbdbjhbdacadfiefhijafgehidjbefggf
caacefiaibibjcccgfehgadaffjbcjajegjcjhi 
#fcdegbjgeddjcgjhdgdfiidjafhbfibhbhjhdha
ahdbhicadhdhebdidhfddhgaibghjifgfbgabge 
#ggecigjbgffhfjeiebdifgaffgdahcdhegjddbb
ejgaigifejjdgadieajejdibbadggfefgfcfich 
#ffbigfigiebbaajgahbfdedigcdggbffbdbegji
dbicbbehbgihejbeheejighebfechcbjgfjfgbf 
#cagafdcjiejeiiejhghjgebcgajcccajaaigheg
edgccaedadbahjhdbgedchfceiagghaagjedcji 
#hbgdhbchejdefdhjaijiefajbiibiahbdcahbcg
ebeceggjigejeciefeahgfigeajcgdigjhhhfib 
#ahjbjeahicciejgbchecibjeaddfcdaigbfebhg
bhfifeaajhfbegadibaihhdiighgdhhgcggacde 
#fhfbjiafgjehdihjghbhjeigeaaichedddejjii
ghacaadfiaeicdfffbdajgahdffibcajajgeihd 
#chcaiaiabibbcjgibejcgdgbihfejibijjihhjc

iaidagjbfeddafhafccegbgeeegjcaeaeifaace 
#fdhdfedfaijhigjdbhaigfccfehcfihcgafcbhi
hbagddjgcffjjiafbagghejedaggibgfcfggcaa 
#dejdaghbacgffjaaajajgcajdbcighcbebahidh
ihhageigibagihfffcidcaiebcicfheeiafddga 
#ahbdfgecbdefcjiabebhgghccjifcfjdbfbihba
ecjdjhgdefjibggfjgbffdacedghgegffafbfbi 
#bfefhbihgjdiegiediaciediahjdibjijfidche
hjgijjcbgfcjagidaejcaaghffhfbjibdjhecgi 
#dihfabfdiccdgicedhgagcgdeafgehjiecgjhhb
hajfgdebddjajehibdddigfdgifhfjgeabjbgeg 
#bdahahgdihicieigafdcajeeahhdhaiebhhccjg
dffiicaghadfcdicjjjjcbabjjcbggfefedacjd 
#gbfbahhcigjeifgiddegidcdfhcdcbahciecjdd
adhddhjcgjacjegecfgdbfbaecfgeegihadggjd 
#ehjcadacfcggicbcfbiaefjabhaheccfdihdffi
dbbfadfebjfdeiidbjgbiagaaicgjfjbdddibic 
#ehjcgjjggahihajigcfbeccbbdedhcffihhejee
bhbheiffaibhjejihddgageaihejcffffhjhgdi 
#gjdifihbbgcbbjbddhbccdcjigafiaaaacfhace
hhdahfifedicbcgadiifidaigifhghjcjjhhejd 
#jhigifaheihgefegihidhcfddebabjdcfihbafd
aebibihghaag                            
----------------------------------------
-------------------------------------   

,the log#aj of #b to #bjjj digits4      
4#cjajbiiieffcihaaiebacgchhidgbddicjbfgf
hahihhadfbajhedacajdbgdfaabgajhahibgddb 
#dejidhfibgbebaahahfagbjdjfhddggaiadcjii
ecgijidgfghhaaccebcejeiiifibcccgjdfieeg 
#ejfdejbifdbedaicdjbffahaigcdcaafjbidcej
aahcijbhihaghehbfagaeddciecahfaibijdfce 
#chhdfiiebjbcicajhdifabdfbedjdjjbfccabei
dfbadghhdehdgcahbhbfgbfhcihbcbfaifedbgi 
#cejgfcacagedhcejibgachifdidfiagghegfhia
hjejgijjjgeiiedhjhghaedeigadehejcaifdhg 
#gfbfabbdibbijhbiaahaijieadihiigagafaihf
jdggfgfejjjfghbjeagiabeegcbhfbhffhcdbjj 
#jdjbibjejihcgjhdegbbbdhiedidbigefbadigj
gbddfeigjhfacfhifjibbaijidhbgfabadciadi 
#febhbceafghbfdibcadhjdjbggdfbdcbddafcca
aechgchbeicjchhcjcichjfccbafacjbcijeahh 
#jehbacaiaefhedfafibijecjaejeacaibfihecg
hdhhdahgahcbjjfegecefidfhcibigagdbacbja 
#jijehifhijhejehefbdfdjihgbahcifhgffdhec
ihefbceafabggcjbfchibghghbfjhdihcffhajc 
#jcjhdcadaeefjhacidcfagfgdedhhefffcdbdec
habcgccicbdbbdfieidcdijfjbabjddejdbifhb 
#gdfjfhhdghedfaaefhdgfhdajfdcgigiejjdfei
fiiaggdefegedjhfdjahdfdjgidefebieddcdaj 

#ggdjhbiciiigdedjjgcgbagjafhjaidhhijeedh
efiajfidjjcgedaafhiifcdaegeibigbahjfddc 
#jchajbhaebjccibchhjhefccaihfhedecihgcic
edhefjfeghdbhifhdhihbfaciddbfjhdffcbghb 
#iebfjbhgffbabgfbcjdcdaibbjbfbhiabaabjhc
fabfjjeehcfhfbedhiiiiijibgidhghdcaigdgd 
#dcchhhfhfbiaaggacaegdacadcbbbhbdafijgbi
iehedgbebffaegjafhcghfecbdhdcggbdhdejad 
#idbcajgjihajegedgfddbciaaaaffidfiadeedf
ecaehbacjhgedegadheiaeebfdjfdffideicigc 
#hgbgdffbfbfdhaeefcgcacecbgbficcgieififh
jbdfbcfcgcehjcgjagjbghfebghgachbcfhbffg 
#dieaihbhhhdfbccfgeegdfbcjfddggiccfdggfi
hjcgadgjfhcaccbehhhahgcacabachfdgdjbifj 
#chghdahcegjfgghdjihifgbicbbcjibbhcfcfdj
ijbjafggjcgafahbgccfibhdedjhgbahjhjaddg 
#gagfbfbeejfiecdgfafjhhfgififbdicgffegec
bjddcddhfhgiecbhibicibeceeaaadfhcagbebb 
#fgbfijbgegddhjfghjbcgfhageecdhcgdjegjdc
hbahabbcbbecccafghifbjgigeeiijcbbicjeig 
#eifgdgbjhfffdhdbcjdagcibcgibeiihfbecdig
ighcjicieegicijehecajcgigebebadcjfhgghh 
#jeeijfagcddhibaiaajijbfjbehbfggccjgegce
eibeghhhdbbhgggibibajcfgecdjghfcdijheec 
#jdgidhiaiedabgdaiahdiieiihdgbjjbhifeabd

hbebbijjgdgfdddfcbcehhdbjhijfejciedieii 
#eheehdiajceaaejdhdibgbahbdjdihjgdeddaee
iigadihidggi                            
----------------------------------------
-------------------------------------   
log7#b71 natural logarithm of #b to     
#bjjj places4                           
4#ficadgahjeeiidecjidagbcbabadehagfefhjg
eejjacdcfjbeebedabjfhjjjidiccicfbaififi 
#dgaefjehfccbfiifdahfhgedbjjadhajbjegjfh
egccfheebjbcegehacjeegjcbfgjgeafcejgeif 
#aicjgbgegjhbhcgadceaijcjgjchfbchiafgcdg
aabccejaaecfddigieebciabjdgeagbfhaegdic 
#bjfeaeeebdgcdaciebehhbiejdecjjgjiecbfcf
ffdbfedajdbciaeghadiebjdcgdjdcjcheejjhj 
#aiddagjfdafgaeahfddgabhciifhagaghdedfie
gjbfbgafcajfdedfaejbegbjgdjbdhafcgggcch 
#ifcheejfiebfjffhcdaacgbgchgcgbbibhiefdi
cedgjbegfbfebjihheificbjaifejeheedgfdgj 
#ccjfgicfeddcbedgfcbgddieabejdjfjfidchad
gajdfhiidfejfbbjafggbjdbdebdebifabfhgid 
#fedfaicafeagdfhacibfgbejdajchjbedfbeife
fhfiaddaibhgafjhbichjcagbgadcfgghbfedhg 
#geffdhejhefgdjggfdhdeadfddciidjdfadbbfj
caicjifgcedjbegdddfjgjcjhjifjhejdgdhffc 

#hebcachahafgfgeadchffgdgffdghijhhadcgad
aihedidbcaeaiigcedhhjcgeafehfabgecebiaf 
#fajjjgajeceehbdihgidadgbiejibicaachigae
eiihbjefedcibhgagjjjgbahjhegfajbebcfhhi 
#bacbddigachicbjcghdcicecjhhggdhbeigjaga
eeiajgjhhbcfhcfbgehihdbehiahececjbdcfcd 
#badcfgjfaahibcfghiaibcgbcadfgbcbagbjecd
jafdibefhgbgdgghbcddececdgfdhaadidahfdb 
#chfggfggddjfjfiefbfegcgifjjhfgjgfbegaii
ahdgcdjbbfeadfbhcgijdhhcjfbjccjfaaddfcj 
#jgcgaidhijjbgdcfdcifejjbehjicfeaiddcjda
aiaaejfjhjidhgicjfghfeaehhgjijjfjebjcdf 
#hdbigcfaichdabhifebeefecifhfjbbaidabbib
dbjgegdcbagegdhijiggjfgebfhgaaehagjeaac 
#gjjiaehidbffedgheieifdhijfecjehdfjbehff
hchbidjjbbhccjjechbjgdjjefggjecjdfghgjj 
#ahdafbdjddahhccbcbgihchfcdijjaefcabahhi
efjfejeecaeabgbaiicihccbjcjgeadjhdbfjia 
#dgijjabfeafhbdcddchicegbdgbghhbjedhfbga
eebgdahggbdcjjbdhigidedjaifahgbccihjhfj 
#hcaffdhaadijicjffgeaiccicabhijdcafdacgj
fhacigggfdihagfigdhfhijchhgghiiiabifejc 
#faibgjgajhhibfdajebcjibdghciagcgcejabbi
hdbdbjdiiefhiceiibbjffjbbjdfedidaeajfac 
#iahghhegddbdeeggeajbjfhcgjcjhfffaidhjhi

fdabahfhjggijbjhahaehhehjjjafhhaaeigcje 
#fahffgfaiiahgciebjjgffgaibadeibbcfgbjfj
becieiedcfedafeecaabieageihiidjjefjjjcf 
#feacefgefijeabdeibfhbegdcidfdhcafhccbfb
dijahjchbdbdjhbdbcadebcjfadjifchjegjjgj 
#beeachggjbfhagheafcjfijbeeacgjcbcdjechj
badejaijaecgdjbiejiidbbfbiieggifdgdbgac 
#haegcfchjagbihgcidjgjdbdbagiigbbffifbig
iicicabgjfid                            
----------------------------------------
-------------------------------------   
log7#b7 squared to #bjjj digits4        
4#dhjdecjaciahbjadbdffgajbebfcbfffdigagc
jeebieaeidedeehfhffhfdaccfbcffechbbeihc 
#ddgbaiiidhbfcddcibfiijicbgaeeigffacehhi
gdhabeeabhdaccehbfhejcaggeeebidhhjhddbi 
#jhciahdffdgihhifdjdccebebdbcfgcfdcfehji
bhhabcjhhfjbifciaabhjgaecjcahbffagfcgif 
#jihfgcjhbgjbdecajeibebbffefcabhbjjbdief
igfdeadcecjgifcjfdjhbijeedhbihefgegbcad 
#igheajaeehfgiajhdifjhcijiaeaiccaajhdhgj
fbdjebhdedcdahbddedibgifgbegafiebiebegg 
#aabceifegcehhjajdaccedhfdgdigjdcebfhfib
chhehccchhbhaffdhgijhhegedjigejiffdigbc 
#aehjejghfgcaigcdejfaddgabjjejicdadeafea

baahfbajibjcejhgdhbjbiheeghfiabgafjibcf 
#dbihfgagccjahddfbdefcbhageifdajhaibffce
hfegghbccibcfigdbhjjadbfeebgifhhbcigieh 
#fiijhaigabiahichbjfiiicjhbehcicdcbccdfd
aedcdjiffcaihgccfhihjiedafijifhhchddffi 
#dgedeeajehcdjghjeedhjdjbfbagbceggcjegfb
hhigfeddccddjhjjehedhigeifbbadaaiiieaib 
#bgjhgidedidfhaciacgjjbcdicbfdicejihfcef
iiidfjajhhfjbihijjfdedfefaicghbajdajgid 
#ighgfjdeggdechhcdcehbfhabjfjgddiiddaefi
agacchaehcijgdbebdghiddehaeagcidhfadbfh 
#fjbigecfdifhiideccefebbiahbagchhhgafije
bffabhbjgjajfjaaiafggcigbeedjbdhjgefbdh 
#hdajageccgejaadffggichjgddaebbjeaejggcg
bhhgefcjjajfdieaieeffjieihbhaihecaiggah 
#dbgbccbjdcehfecehbfgcjcjhaiibebgceejbhb
ghbdgehceigejbajeabjbaefibefiehjgbicaad 
#gaiefacbcdhjcggcbebhibigacggafgccijbihj
bdjhejbhjffafehbjjafabjhijbdgdecddefdjh 
#jbjgjhciaefddffcidjgdbeaghbbehbcgcibfgc
jffciccdijhhfcjidadgbaieadbjbcchiehfgde 
#fhgaeehjdddiaehceghcfejcdajiiechiedigij
gffgfhhcbbejagfbcjeigeagcicfggdgaaeigai 
#ddadgijhciecdfffcgijbjegacejjfdgagchbba
ggajjcfefbfbgejebdgbegdjabeahaaccebjdac 

#ibgdicaadafgiijehaaidaehbaifcdacagcgeej
eeadiggdjgafhfaieaacbdeiadeafgdcjhbdbbf 
#hibibjchjdhceegeddgeadjdjejdfajbiiaejib
ccfjhcadfbjghbdfjhbjaaggdjebgaecjchfdad 
#jdgeajeifdcbachhjaigcddahdgiefbbjiajdjc
dibddaihadbjiagdibhfedffdbhfiebahgbghgc 
#jfidfbiiaghbeifecegjcaaaifbgdfjachijffb
gchficjgbjiefcgibcdeifcfifccbheddhfdgbb 
#gaciddfbijfdjajgaddhdeageheabcaahfheehg
bgcjbecaaeeacbdhchheiageifagbadbieghajb 
#gjeaghhjcgjjggdbdidhbaddcejjhaedgijbgbd
jjchdigcecgbbcdafijcdghiicbfcbafebeeiec 
#ehjjgidebhjfgjedheadicdfibjdeegidhbhfeb
hciedbdcgfjd                            
----------------------------------------
-------------------------------------   
log7#b99,pi7 to #bjjj places4           
#a4hcghggjffdjicdedhcefjfeidgbhaabcebgig
bbgididgbgeeffhbefcdcjcjhjifeecaciahede 
#bjgiechidhfeigbgaijhciebddjaabicbdibfhf
gdhibgccgbegfcfhaehgaddcaageahcjddecfbg 
#hgbjgabadhejidgagcchjibgiahaaihbgfafaab
fjcbfdfigdfahibedgdibeajcfejcchiijhiedh 
#bjaiagahgjbghcifcbbcaifbfaadhjajfiecijg
gbabiiagihddfbdbgiaacheedhfiiidbbjjefgj 

#ciaiffchihejfbghhedabibeiacgbidhhbcabdi
ebdbfjigdgcfcjefhiihgehfhhgfdffjgigjbeh 
#iecjicadefchfcdgeigegjfagacghhdfbgbefdc
jgidfafgbjebiejehecjihbihjjghgaaaiiiiib 
#jgdabfidcgjeaddjdgaebdcjgjjfhgbdgeibjed
cafigejjigbbgaijgfhdifbfehcehbdheciiibb 
#gecfgibhjcjbghiegedeiajjbjbjffdagfhcicf
gabchhaeieadccbebedaagejejgfdigbdeahfje 
#jeijdbafjiijcfbdjcicfajdeaifjjiagfajgga
digfgjfehhhbbghacfaefeeeecdgedddejgfbff 
#gfeahgijadhbhjdjebchfghgdbfccgdjhiddacg
aahiaefhfihbfeebjhaeijhbfjaecfgifjidjce 
#jeaggdifahggagdiaaddfdfejffhgghdhicheei
feegdiicgjedbbeafageafbccagdhgejehjagfi 
#fhiffahcejgghhaebeiaijhhaihificegifjghc
bdbfahaddfegjbhgcegbijgeabdgeidbjgjhfij 
#hebfcdgeegebibcddjgbbbhcdebgeceicgfgiac
bchjedjadhhbfjiehbbhbgiiigfibegfabaghab 
#gbcegdjiaedhjijjhhheibjjjacgbaghjfgaggd
idibdafaggeieijdcheficgbhfegchecdeedeaj 
#hehbijaffaefahieddbigbheejafagdhijegaga
beadegiffcgfdebdbcbfdbcdbaahbghcjbgebgi 
#cdeggdajajgdeffbceicihbiicadfaajcibjchd
gbajdcejjgdgdecaihegjbihjbffbbhfdieehhb 
#jcfdjfhaaefadjehabcgfigchbedcgaahheiiei

gceffdfbhedeajficaacbahcjjaachfcidfecib 
#cjfjejgieceahafgibecchaiffcbihecdggihhd
jcfjcgjbjdghacefjfdcfdifdggbiijbgfjdjjb 
#jibjabejfabccecdbehebijbgdifidjadaiaiie
eeibgidacccihgehfgjccacddgibcahhdjhddec 
#cjidfcfjgiigadhehdgijagceeecdgjbfcjbega
jbdjbbbfabjffcdcadhbeijhehdbhgibcgigdcb 
#djdhbeiejdgcedidibdgfeiiefjiaedcfdaefff
ijjbgagbfebbbdcajhgjggfbijgaiahiiahgdjf 
#bhahcffgacigbifegigiiidfigdjcbbbfbbehdb
dhcjabjcddhhdddcbfiefbhffbadifijgbcafbd 
#dhfcfhhcigggejifejhahcfjdbehajbgigdffhg
aahcbcgcfcjafhbeccgbgaefjbiiicgihhehcfh 
#cihijceihaiahagiedgadbcbhaiejbdgicjjhde
ccafgjcccajjacgjgjffcijbigdhbhccgjbajej 
#ejhigjifheggbghdhaedcedihfidjhdfdeifiae
gbabddeaijdcbfccfiadjhhfaeidbcijighgiac 
#bfegffbabijedhhjfjbjcfaddacbddffbjgbafd
fhbfbachihhb                            
----------------------------------------
-------------------------------------   
log7#c71 natural logarithm of #c to     
#bjjj places4                           
#a4jihfabbhhffhajifiaciebdebcfibbebegjdf
dgdijeeghbbgdideagcdfidcccfcgdidbicbahf 

#jhiffhgcfaegedhacgcbjhhghgigjjbijfeiegh
fegdbcfhjjdbbeicjeaihbajebhjahgjgfgbggd 
#ajfjcafbgfiahcchacfgagicgcfihhddcfjieii
jcgdbegjcafgieiaaebaadeeiaiaggejfgacdgj 
#edidjaffggeehjbbbbjcagjbebidfhigefjfija
jfebaejefdbhfhachjcfcagcgcbiheggghbcffi 
#iafedgibacahahadijbjjcjajchbcfcjabbbdhf
ebgdhaihbbeiiajigdebdijhifdehjecdfgjjhh 
#deifejhegdhdddaaijahhegjhgfdgdidhfgjgif
acjhehbidaafjbaffabaahdjjadjihbeeadciai 
#dhgfhhicfgihdidcjbbeegcaececbifhecdebie
beadeibachgfdidfheicbefbgiddafeefidaegh 
#bgbcajceeafhhffajbaahdfihijdciidcjfcach
beebhegcfdffhhbhbdihhacfhbbhjjfcdadciaj 
#ghfhicbeadefdcgeajbjddeafbgefaicdigcihb
aafidaehegdjececfagehijjigeabbbccgiggcf 
#ififhggedcedgieacegabihbaggjagehabdbabb
ceadjehajafcbgbdfeehhicgbdiefdiaiahebdb 
#ifjgiffhdbcdfdgjficggbcgbebfeejhbjcbjgh
cccibhjeehibhecadfhgcjieacbfjfdehcjiahd 
#cigdifhbbbcjcbegfedfgecccaahbcjaifdibge
begeiiacbbagheaceccijbcgdhbifdcciejbedf 
#jgdbdehbdicdfffhffabahhadcfebfefedbiedb
gfgfajejedgggiedbbiccigccbcdjaagcgdcaic 
#igdegihdgjaheeiedhdidjeidghcecidchdajaj

fjbicjgfbbibbbhacabjgdhicjfcddecdjbebgg 
#gcbfhefbgadhjjafhahgaedgbdcighbjgahghjc
dddfghjbafaghaehdaijdbhbjjgfgbabdcbeegc 
#hjadcfdaghhgfhbfafajdajafhahgbdbdjfhgij
hijiibihgdbjhaebahcbcgebhidbgebgcbecdjg 
#ajjbhcegejfiejfbdjcifedfbgebbddcjhdfbeh
hdejheighfbecjhcbbdggdechhhejfhjjcdhhcb 
#dcdjdijjhciijjehjhjidcefebhbabbcgjchhgj
bjcfhjdedhfjjggfbadbddjhhfigbeidacehdcf 
#eiiibbfbaagcifgjhjdiejiebgibgadcfcaedfd
jdddfbcjhiaehahecfgaaifjhcgjcjdhecebjij 
#ifgbfbiehbdaejdjceeiieabaceedejccbbdagd
hdgdajjccaihadhghcbdebeficcdgjdidiicgcj 
#afefccfffjiiaijciegabbhbbhddhhafgdcabae
jfbhefiiichgdjchhaijabgddhciefdgiajcdgg 
#bhheigbaaihejfdidbbgifihegiaffiiefdahee
abfejdaejbaiaeedgaiffehefibigbffjfebceg 
#cbicgcfhcjjbghcjibaggffjechgjcjdfbjjgff
aehdidfgjjbbfjaagefgigeahjjcicdgiagfcbg 
#ghddiceadbfcdifhcfjjcgeeghegafjgjjdihah
aeaiahdcgcdchbijicdgdfffjdeggejfeibgcfg 
#jabaaaecgjehbdifdgihdgicjdjdbjehbcifdge
cheghejifjfbfjicchiiadgjfabjacjbdcajhbf 
#jeahbfbiehfdjjgfjjcjeididcbaaffhhjddfaj
facdfhdeccih                            

----------------------------------------
-------------------------------------   
log7#d7_/log7#c7 to #ajbd places4       
#a4bfaheiejgadbiadhgdaiijedbbhfheebagjhe
iiagabhjbfcgfjheegdacjihhgfggcgjdjbgfah 
#bifajabbcdecggjihijcdiaabbgjhjcahgffbgd
cjchihdfhihbichgbiejhbgcgbcibghffiiijjj 
#gaiccbhaafidhffbccedijddcabbeaibciigjcg
cgcdeehegjhfhafiijfbafbdagfhchgebahehjc 
#fhcgaiahgfddcgdjfafdjegigaeheacgehahjbf
bfbfeeaedcgefdigiejigiebbhgfjjicihgbdgc 
#bcjgaedjcjcaeiafeafddjfbchdcbagjechdigb
aefiggaahhddjaijhcbgbhhdhghidahhahddcjg 
#iaicdeiijbehidjfcbcbhjceihcbfccjaaacbae
fechicihbaefghfgbdccidjiffdhefjccichfha 
#ffaghgbdbehaihiicgddhfdacgfbaaagjffebij
cfddgicicejbdcdjbbiafjbgchbgbjdgbgebcgd 
#hjhifefhdhicfahfidcfahadcdfdfefhchedjhb
jcceifiiebdjdjabjijafcjfaehhbghdgdjedii 
#bfajhbigbfgeajibfaagejjejbjfejihcjdiaee
biiddbhhfcghfeadgjfagfgaggaejiijcegciib 
#bjibggfhjagcaiagiajdcdacgiebcgdcdibbgca
degcicacjeebeiehcfaddejhbcfghjeicejdggb 
#djedeacfadbjhhihgjhhcebjgadfdhhcfhiaegg
jbjcajjhfaiechheihgieeeeigiifjcgdhiidfe 

#aeiaeiaahgabcgaafgajhejcgjhfiigdcigfdca
gecfheeedeabfghdijhafhbfbhjhejegjegjjij 
#bidfcdbjgfieibffjhejjfbcehadfeagfcddiac
fdahddjfcahebgbjejijicdhhhibhggbdegifag 
#aigdfdifjig                            
----------------------------------------
-------------------------------------   
-log7gamma7 to #ajbd digits4            
#j4ediecicabihafddhbbccgffagfhhjbijgghhc
cjfihihabfcjfdgiajijaeacjde             
#gffcadbjjeegecjdgefbfahihiaabgfadjfhdad
ffibgegiaijdjdieebfcaheij               
#edejdaggcdedihdhjghbjgabiciebefbhggbeji
dgjchgfbhafhhbgjigefajjbf               
#aaefdjhhgehhfhhidaiaciigjdbfecghcicbcea
jhcbcgdiecfhgadagibihggid               
#cjjjfihhjcaebijfjjfhcfcibceeahfhaichdgg
ceehchjeeidafhcijhfehfbej               
#bhdaeecjfjgecijgcddcdbfjgijifejfbeiacjc
jfdeicdbcjjfedabcfajfjjda               
#aahfjgdhibagdffgejfdfaebfajjhdcdbaabeih
hbdgedfajhhbchhbdbhecidca               
#fdedbhaehiejfbijicgbheefdjfcagabhfjegcb
hbfjafadbiaejbfcddihgdgij               
#ifiaefgfgfjbchifejffaihibbedbhfjagbfeee

facgffhbcbhgiigiajfgfccif               
#bggfijchbfcbgjhibaihcidegbcjjdacddcfahf
edjgieccfbcchgbjbdegaafdf               
#jfiaddbechghfhcahjfeihaddagiajfbgcbcbhg
cbihehidcehdjddfdeacefiij               
#fhcejfgdiehfbifejgbdggjeefidejiihcbibje
fbhhhefdhbcbhichhgjadchjf               
#dedhbifcahejfbgegaidfjiegaeifcaifjihccb
cieijhjdchadgbgibdbjbjddi               
#eafcjiifgfadfebgbeaidgegjhahgdhecfjjjah
jedabhgjfcjafajehhfefjaji               
#gjaeeibdcigfgffiihijidbeeibifhahgjgbiaf
jjgfghijjhciciibghagidcjc               
#dgfcijceadbjchegcjhgcfdcecghdcdgdgagjce
ghdajfiegceigfjbjgbcdedag               
----------------------------------------
-------------------------------------   
,the log of the log of #b to #bjjj      
digits1 absolute value4                 
4#cffeabibjehaffdcbgjabdciaehbcbffidfide
dbfcddghcgajebfcjecfgggacfgjefafaecaice 
#bgchedideehbbheffihijhcehcjbebcjdecfdhc
dgfeeffcdbeagaidjfdffcdhadfeejcjefbgiba 
#chgcjbeefahibbfiigagfgbbhfjdahjjhcbfacj
aiidbahiecbheedfccichijdfaehcacbhjfcgaj 

#djhjicfichddagaadcifgdghdfhiiagjebfdbbf
gdahdjbgciajcejjdfcaaijdbbahicbcfhjdhdj 
#ieggbbgfhgedcehjceeiffdhgaadiahhdeehacb
jbcccafcifeigfhjhbdiicfdebiccehdffhdfbc 
#egdbfcdcgefebecfihfajiefiebaeiaggcegbja
jfejacdbbbbddiiidaecejgjghadbbighccjbif 
#cgggdcaaigjbicigdcejcfbfccaefbddfhgfeej
hhfjhiidacedijgjiijhidcjcididehbeffhddi 
#dfhaaebbgagjfcaidcbedhijddfigefdicfbbcd
fihaibgjdffgbbhjdajjijbifahhfhhjbahbjag 
#diajaiibcfajeedijececbdhggehecciiihjigj
gdcggiafbghbahaicddaiifcaijbajejdbiccja 
#bdbdaejeibgibfhjigedjefadjbibjiiagfaehb
bdhjbcdbjijahhdabgcfhabbficbabhdjcdiccj 
#ebejggbhijiafgegegiacicehhfbffgaijhghfd
fhaaeieffcieiddbiigiehjbjfciedhacieibcj 
#bgicghfgdficaeeibjbgaiccjicbbdaeabdjjgg
cagafbdgjehiicfijgjchbgecceacgcibcahjjg 
#jajdaieegbbacjdefccbehffbhidgcbfjgjahdh
hbcciaajjcidajgcbhjffecdjadiidfiegcjbaj 
#hbgdhbgdahigaebhgebecicjcjfhhachijgbiac
iebjaabhddbjhcjcdcbchejdhffhegcfehghfja 
#aijdfebhdcdgidaihaieiechebgeafjgegadhhd
ghggdebgjgabeagecfadiahdgaaeeecgdehjicc 
#ejadgfcddheefhihbbcegcbdbahhcjeegihfhia

ajhjhceeajceagedbfegcfihcfabfdcbbjbfcdc 
#aaebiieidbbghhjcfbaghgefcbajcibchdjadge
caafdadaiciahhheejbiijfccibaidjaighfchj 
#fiigaedfdjfgbdbebjidiabjjbdbffediieddcf
bdidffhehafjjdjihaeidbagdcfhhechfeabjba 
#fjdjgbbfidbdahadhcjjjiaefhgcjeafdchgfid
gcgbhaijaiceiafidadjhdhjjceghbijgcecdhh 
#cjgfaaaheejajjjfagddacbjgahfecfjhfhabfg
iggdajdbfddehjhagididhdgjjdchdejdjjiccj 
#djeeceadjfadggeadejchjdhcjcfadgfcfcjcah
dgiceifgchadbbcbcjijbaahajghjjaggbgdbfd 
#bajhahfefabbifiahgcfiaahfhichbjejecfjdb
cidbfihaeefjaggebecdiagaadbacbheeidijcd 
#ajjeechhcgiehihcbgiahjajbfjjidjadccccbe
bhcheifiidjhabcgaahfjjgdbhdbbfhdgjdacde 
#jdicdeghibcehgdabjjheidijgcfbfhgahicegc
ajiaajhdhcfcagiebfiagjggfiejhejffibcgac 
#fifbdcjehhfbjhfhjefacjbhifecbfjdhjafabg
cejiajjbjiddifefbiediefjjahddgchgbchdef 
#ffhdffiihhcjfhccgcdghdibdcjicajebcgebbd
jjghibgbaffe                            
----------------------------------------
-------------------------------------   
#a_/b                                   
ln7#a_/b  #a_/b #e   7 and here is #bjjj

 digits of it                           
4#dhabaahbejeifjcddgdiggehiacdbdcfhdbcac
eahdccdcheffjeaiffajahafhhdjafchfgfjhbb 
#aggddabjjidbiabbgbcdgdiigbcahciiehbicfe
fdaabgbefhcbcgbfgcgfbbgecjeibdahfddjige 
#dahbdagjjgbaahcgaejbbchbcicgdfiahgbgebd
cbgiaicjahgigjgijjcefagbfgifiddedegebcj 
#ecdedcdahhgfebheecbefdijbjgciificdiffah
geefcjajbabciifcfgicjhbjfceiiggihhejiih 
#jaefhbegighebfdicbhffffeaaafbdagachjhbg
beibghhdgijbfjifeccaacbdgbbgeadicadjfdi 
#hejhhicbagfcffjjbeffffaiecbajfgifhagegf
fahdgcjgceaeihfjcihdhdegededabjefcbcdac 
#egjjdghjjfcidhgbbdcaebfaghifhjjdejicfci
jebejcdijdghedccebaighfecgjdcgaicijcceg 
#fggbdafgjcgjdagfdagfgighjcaifebcbjiifef
gehgiedbafacageiighhegdageihhcjfibebcii 
#gageijjdfdecifjeegeeabedfibifhhjgijccfg
jdifbacefbiddibeeeabjchcicaggdfigfedhbf 
#iggedaijijjbadhbhgeiagiejajdajcaejjiagb
jdjbheahaigfcjahhccdcejgeiicjeejgehabfg 
#dbacacjcbicdiiajggchhgffgeaghjceacebchg
egfejhgefffejigebaaaeaibejihjecbeafaggb 
#ccedadifijeaaiajcacgfjjjhbihaegecbcifdd
fjiicacfaaaghcieedifebcgcccghjfbddeaaeh 

#igbechechabefbecbddfgajecibgefbccfibhaa
iffecggifaigehiagfffgaajiehdfcgcfceihde 
#eigdcgaceidceibjecdhijgegbfbfacdehbaded
cbgfeiafggiijhjfbibdcigbgcadfhefeecfcad 
#jibcaaciahiejfcajhecififfcgeggebgeaajgi
egggjeaggabdhdbgdidcgdjihbcdejabfgiacfd 
#iehbaifhahhhchaaaefjagajifgabccceechccc
icibgjbgejjiifgbjdidciagccfhhiciegiifgd 
#acjgfceiggbhcbfjieajagefcfdabjchgfhjjag
bbiegdedegbgjhjafbhfigbfaidbiccgbicdiff 
#edhahgbegjhdbbgjgcffgajhaeaadcbghgdjahd
aaeccbicaddeacdgdggdgeaiegjjdgiigdfdfgf 
#egcfdadhcfajdidijcbhjijbbdiggfbbegjbihf
bcjigcfjhehjhdebgeicegicdjfbbgbcfafjgea 
#igajhhdfaihdejbcajeeiechjeacafdbahfcidg
bciheididfdebcdjgjbcdjdfieechcefebafchh 
#hhbdjggbiifbahciejfjcbifecficfjajfbhcaj
fjjihcihjbdihdbfjcabhaidfjgdjdaiedfhfgh 
#fihgedbhiihjbejajhgeagbfbiecgjhgbhbiahd
hcadaijabiijaeaiedigehacdfbjdiehafaeegj 
#gfecfbgedjhdeeehbefifjabbabagcdifgdajhj
djdjeaighdeaiaejgaeabhadiajahbbedfgeabi 
#fbdhehehabbbgbbbeeihhefaaabajgibicebgbi
jfijgbabhhbfhijiihhbegfhhegiggejcjgjffe 
#efjedjdiajiadiiajiciejjbdafbdibecfjbigj

hgcbihegffahjaahfbaajdhijaejhhafdihdggg 
#dgjfifgcidgdaafdeccdcddddfcdjdgjfijjjfd
ihdjhbcggajb                            
----------------------------------------
-------------------------------------   
log7,pi7 natural logarithm of ,pi to    
#bjjj places4                           
#a4addgbihhehdidjjagdadcdbgceacecjehgaaf
dgbidhabiaecaaegaeacfbcjgadgbacggfihhdh 
#bfjgighcfbcbgjbgedhigjggjbjjihabbbhfigi
hiaeijdhbjeebgibcdefehgbgijhajghhajbhfh 
#bebgfciciadbffcdeijbijbdhdggccehhfiicgg
hibjcaaifcjhbdgefgidjaaiafjbhbagbbgcgih 
#hhabfefcaghjdihbcfigcaccajfiejjcfjjjfdd
jedhgbfchhjbbcbgjjifdccejdieieaahaejffb 
#cgbebdfhcdcciabfihifegigeadjdgggjcheggi
ieciihbehdbeffjbbhdhejadhacfbagiaeibebe 
#jefgjgfchfhfjbhjgfcdefhhigejeabccdcfjgh
adciiadaddbfdbieifgabhigghaacfebfdebcde 
#jdajeijjgafjhahegjhbdihaahhahcahfhigfgb
hdeibhaajbegfefhgeagbdbbcchccgahibgcjdc 
#bhhbagcdhfeajdbgfaecbcgeafajbhcibbbacdj
adcfifgagehefafddbdgcgahghjejfjdffibjef 
#bhccggcajaccfbafbgdeaehihgebjaeabiifede
dfegcifiaebhbebciafiehebdecgiceidfjadjj 

#cgiiefeaifffjcfechjjjaabfeihehejjabigfe
fiijfjgddffgdeedgbfgajdejhdiejffheehgdc 
#cijggdbeacdaeibdabfebcagggaghdiaggiiehh
jiegfghhjeajbifdddgejijaejhiaadjcbghjhj 
#gfhccgccgichididhhjgeaebhijjiahgecfcgff
jhfgjgdcehcccdeajhacibcbeceegdjfgfhdcbg 
#dcaaihjdifcciiigfahjcjdfbbabhfcfaeiehei
hcfdjdgehjjihfagiiichbfdfbibggfdfbgeidh 
#dhdhifdadajgdhcacbeicdfbjecfcejgcjdfjee
jcjgfhbaediddddaedgghhhdeeiecebbhddjjdg 
#hejiahbagbeeiaeagiijjghebdcebchcgaabhfg
acbcdbijeeffifddibeheehbfbcaahhbdbbcbdd 
#ffadgfgciacfaecccidadbfdecdfjjhhaigiaee
dghifggegebihghcjgeicbcjdiigeagjfghecgj 
#fffcaebbbacdgeajbfdagcbdiahijfecdbegcgc
jeahcebbhcafggfhggcaaddbiddcfhajhiigebb 
#bhgfcdeedijiiccdfibecihajbhcihcghdfgfie
jgiigaifeafcjjhchfdifffcbgdbbchhfgfacib 
#iddaabcgifjfebijhadfcedeejbdaeaicfdccfh
djdjjebbefaeegefahjecfhjdeifacafjfhfcfg 
#bbfbigabfhdhjeeeahjchbcifbdjegihcachdcc
ieehhbdhceefhaffagccijahaieejhibdffgghb 
#jdbhihhgichdfbcjhaiecejgjhbebcfiijfeedc
iafjbifgfefecdiejidhgajbfhfgbfdjejcfcdd 
#hhiieghaciedhdjhjdfighghfjcgbcefjjcajcc

eahhijaffdajedbbdeadjdjjhbadhjjbfjgahic 
#ibdejbjggghefcefihhajficbccffdcegcgidha
jibibgghaicfbfeihjfadbjdbgjjidcihbihcfd 
#gccgfgibbejacjedieddeigechjjcgfdgfageai
jhbfebbidheggbhhbhcdicgiiacdahfihifdjdc 
#dhcdegjiaeejdfjfbiiabheifadbgadcbbefcgg
fiigidcbhhhiebcjgdjdadfcebidffaaccacfda 
#hhdaibegdhhhahicbjgifegaiiadddicidjbecd
hhcbbhbfbhac                            
----------------------------------------
-------------------------------------   
,the ,madelung constant  7in absolute   
value74  7for the ,na,cl7  ,references 3
  ,richard ,e4 ,crandall1 ,topics in    
,advanced ,scientific ,computation1     
,springer 1 ,telos books1 #aiif4 pages  
#gc-gi4  ,andre ,hautot1 ,new           
applications of ,poisson's summation    
formula1 ,j of ,phys1 ,a vol4 #h #f1    
#aige pp #hec-hfb4  ,david ,h4 ,bailey1 
personal communication1                 
#a4gdgefdeidfccahbaijfcfbabjceeddcigdjcd
heafadcffbdgdagehaebh                   
#becejgfejdjfbcecbgfaagihijgehcfbfidfjgh
hiicjhcbehaechgecgajeicbhbjbiiddahchbhj 

#acjcficcjjbaefeiicfcbhbcgffjgagbbigefhf
eibchjcgafgbjchajdajfjcdbadeefjfdchbggg 
#ghfhcbagcacbbdcfigeehggcdbfbejdgdghghba
bhejhfjefgiaffhafgegciibddgfhdabigjcfgh 
#beahegfbhajicgacaccgbjgfgjgaicaigdbdiga
ehaaegbcjifiibcjiffibgcidifegghaajgbbbf 
#gaebjedgdjijaaejfhiaegafehcjhbhbjjejahd
hibaaghjcacdfgcabbifdihehbhhbhahdcegacc 
#aeiadcagjjediefcbeccdhhgecfcjbfgjdbefbg
dhfidhdchjjbhjjbeibgjjbfhdgeegdcfdigeej 
#dibbdfacfbciibjdjjaegejfcjcigbadffdhaaa
eabcgcfdjajbiejffjaaicijdfgaidcgccabecj 
#ddeajbiaaeadfcigeiccaiahjdgiggidfjiiccc
gdfdbidbfefbijhificddggibifhhedaijddjgi 
#adbeehcbgbaiigahdjijfgdfhjbcgfaechicedd
efeejcfjbgcjbheddjhdicddcjbhjfbfgjddahb 
#dabjjdcigdahfgffaggbddgefciecdddbcjfhec
hdiebgidcehjgeahiedijcjicjejgchdcieddfd 
#bjfdchgagibfcijghjcibjgddbhbjigiegiaggc
fiibcjdjhbbadcgdfdeffhjdcajefibffcaigee 
#jdeibbddcbdhjgdhidjhjfbdgdicfajgjicfcji
adibbdcfhihficcadjijcgifhbcbdjgijjdfbhd 
#dhgcid                                 
----------------------------------------
-------------------------------------   

,the gamma function has a minumum at    
this point4                             
#a4dfafcbaddifhcfbcdabfbfeiedbcbegbacbhd
fhaifbjdjjfddfceabieihhdji   is the     
solution of the equation 3              
,psi7x799,,gamma7x7#j  the point y of   
that function is                        
#j4hhefjcaiddajhhhgjjbghhaeijjehbehhgccb
jgieaeccffiijcddhhgabjjafei             
----------------------------------------
-------------------------------------   
,minimal y of ,,gamma7x71  ,the gamma   
function has a minumum at this point4   
#a4dfafcbaddifhcfbcdabfbfeiedbcbegbacbhd
fhaifbjdjjfddfceabieihhdji   is the     
solution of the exquation 3             
,psi7x799,,gamma7x7#j  the point y of   
that function is                        
#j4hhefjcaiddajhhhgjjbghhaeijjehbehhgccb
jgieaeccffiijcddhhgabjjafei             
----------------------------------------
-------------------------------------   
,bessel,i7#a1b7_/;,bessel,i7#j1b72      
#j4figggdfegifdjjgihbjjfgijeibeeagebeiid
hffehbfbiihjbabcbcfhfcjjhbh             

#afecjhebgfdfdaaabiififefedahbfgfefhgbci
hbhbahggcifdaccicaacaibbi               
#faaiecbehcidhbfgaedjbccfhegbjggjhdfhgic
afecbeifgfhjbfjifiicddggc               
#ebgiacdhjgcibhffibedgbhgghhibficdafcacb
eafcedacfjibbceafidiajggg               
#ffgabgjaigihiiaghijdceeabiihbbgdhhdgdag
haeaahehbhbgdgdcabhjjjafh               
#hcigcegejcaehifcjeehadhdefgbbhabggcghec
achicecgifdegdidiaaaddcii               
#egcidifededjhfdadijbdddjgdcifehdfbchcdj
eaiafihbadfegjgededaebcef               
#afaighibggjbaegjaiihjhddaecbefidhdcbdgb
jeecbjdchbedfjajhiecfieci               
#efgefadajhfageieafacaechbjgcbicacfddcij
eaaeghhiiaciigidaahdecagj               
#gbeedccbadbddcagdjdgecbhbchgdfhbcbidigg
hfjjiageibecahhefjaibaggd               
#gddiagcjbdgehbgeajehhecjjcigiihiaifadbh
fggbihhgbibjbfiaahdgigbee               
#daehddhiajchcbfecbdfbfaejfjbfieiehjciea
gacbecceeahbijecihfdbfaaf               
#jabbdadehhbfabddceahbejbbeeichibfcacgej
acabgdgjiffgdijebffgedjif               
#ecfjdjbebdejhchedhhcehidjafdaaeefcfdhcd

jgcdedigadjgfcfhhbgbifbff               
#eacdbdfdfbdccbehcddeicajcggafbgiigfdedi
dafbabiebiabfffjhagighdcj               
#jchaighggedeegfajfcaiibdfggaccaihhjijea
jbbgbhbhhahbdgehcabjajfaf               
----------------------------------------
-------------------------------------   
,the omega constant or ,w7#a74          
#j4efgadcbijdjighchgbiiiifhffbbajceeedig
echaeghgahfeabejhaceacajgib             
#bcjdegicjhffhdeffficbaiddfifagebbideegf
chjbdigbhffghighedebcehdf               
#eidjjgbiiefjheafdcibhiiidfadcaaegadibie
ihjceidcgffihdgdfcefjfacd               
#bbfhdfacefihiegjdeciggfbdheegjghfehggcc
gjfceffcccjabchdcjdeefced               
#bighfjhejijaedbijhaibjhefjeegebcgdhaife
hdfeiejhjgbgcjhijejaegccf               
#ahcaeifjgjffgajhjcibhciahcfjadidiifdfcd
icdhddhcagdfeiaeiccfcfhic               
#cfhjigadijhefihcgageajjicedfgibaffgdgee
bhhigcadgeehhibejcjegbhbb               
#dfjdhfeabdhedajifhhcahddhggjdccdfggbgja
fegddfdgfebjjfbgjaccfjdid               
#hjeghhchgeggdiadfceihcjcdhjhfhfihefbgcd

bjiiaeaaihcjfacjbejbgjbbc               
#gbibhchgbgbafedbdbfegbfihdcjfichijfhehg
dhbifdbafghbcdbeejdbjjcjg               
#aiffgiebbjhieeijicfiaccdciejjeaaedidied
bgafgfhghididgjbddchcjccg               
#hghdjjbfjffcgfjijihefcfdehbhghghahgiecc
hcjdbcgdgeeeefifigedbhffe               
#facedhjedjjijaajdggabcgcbdgcjafhjhhdbjj
iccdbeiaiccgdcjaicedffbce               
#cjgfefgbgjigegfabahhdacheiidddbhbhjjbfc
ebejfhdddgebeeifbbejfaach               
#ghdhabhighdbgfichhjdgbibjbfhhhibcheafdh
dgecdbchddiecijbghifjiiga               
#ajfjedghdbbabjfacfaihcaaaigccgfbbgigfji
fdiajaaggajhhacgdjgjdigcb               
----------------------------------------
-------------------------------------   
#a_/7one-ninth constant7                
#j4ajgfeciaibbfdhdegffaecbcddejijidgaije
hgig                                    
----------------------------------------
-------------------------------------   
,the ,parking or ,renyi constant4       
#j4gdgeigibjbecdaadceaghgcjidcfcfebdbajb
fagb                                    

----------------------------------------
-------------------------------------   
,pi_/#b99sqrt7#c7 to #bjjj digits4      
#b4gbjfiijdfceacbfggehiaaagchfdfcbcceihd
bfjiicgbaciaajhfccedhbgdjcjhbahdggafhie 
#cjhbeebfahgdhbcahjijbecbhdcccfbagbaeiig
hhcefaidjghgdajeafabbbdfigcghjdgaideibh 
#jgdbgbaeaajejgafjfjfiibfdjgcbjfcabaejej
bfjaiidcfdijdaagchfaaiidegcajdfcdefdjig 
#bgfbeadbedjjeibccggiachdhfgdifgibgcheca
ejcgeeijhfhhbidaeicibicjhagcejjjfeecdcd 
#ajecefaafgebgfihibeddafbachabjgfdgehicb
fhjdjjifjddhafjegddiacjieifhjcdhjeheedc 
#ecdigjghadcgeeggigecjegcajdibdiijhdbagd
cjeieecfecdaibbjjedebhjcafafiijbeigjgga 
#jbeebaagcdjhjabecgihdbgbaaddedfidhidjcg
eheiiaejhfhfbafhcaegfjbajchjdheadajcaia 
#efcejdjbjdjaahhedhcaddabbhbjcebbdaaeihh
hbehceedegagdbifiacbeecafjjcgfeaciaiefa 
#jaigajjcghgihcfiecehchgiijaihccibifffcd
ghaeajjfbejjgejgefiddfijecccccgjgcdhajj 
#jcibbbcadgafcdfebdhhegaeeeahedhbejbddhe
fiajcgegifhefgificjchcfigjghhegbeaggajd 
#ghcbdgehhffaaecddddfcdbgchgheibabhcbdbi
bgibdgddhdajafiihceabjcebbhdhgdcajhfidf 

#ceihjhefdhehaeighgfheafbhgiahfafddccgec
ccdegfhadfeihhgehacjidjieehhhgjfdbdfhfa 
#dbabeiejcchcjbjihcjbcagddcfdedbbeicfehf
fahaehfcfigibhdebaiejigjfcjgdfdaiabaijh 
#ibcjhabjaedbgefcghcjibihiifbbiddgaheehj
hhabfdeiaecidfihehdehgdjffafeaejiadbech 
#gidgeedjdfbgbdjagcjhjchgeibcgefiigghbje
cbjgacifbefggaebaidgjegbdicheggjacghjee 
#feafabccjjhiajdfbaefbeceejcbicgifcahgdi
bbjcddbehbieijeafhhbidchcdhdiggibdbejfi 
#ifebcbgihcieajaafjhhgaafgbeadccfadgjfdc
eghjbfchaaebiifegbbdadhhhibdfhbeiihbjdg 
#gaicfhjabagcahhijjffbidfejifeeigdeihjdh
cdfebhigifhfigfiehhhhhjdjgfbjicfaaffedj 
#gjdcgafjidgacfefehjhecdjifhjcigacfjdchf
dighjbhgegeggfedheheadefijbfibicjagheec 
#ebfccfggjbicchfijabbaebbhcahihbjdedfdad
agjgfhbighjhbjhdjcgbcfhdbjdbaedbgdjacaf 
#ijbedigdjjhbjfjcifafaiceddajjajjjjdbcbg
bfjhcebdafbgajjfiaedbahcficigjbchgifgfg 
#dbagjfjcagicdaiefecdedajdfiedijbgcgcaai
bachacjbfdjfeggcgihjdghejgcgfgdbideahjd 
#fffccgfigacfidgejbchhiiajbeajgffhbidfic
gjjbgaeibdgdbgchihhddjebdegaddaeccdbafg 
#chehjjiaafaggfhjgdbideefejjdejcgdfghbei

iiejjjbhjbbfceeafacebadcbeifjjfddgafghc 
#bafhfejbadffjacijhfbieeihcfjjgfbjcjadbc
hfaaifdcfddeffdehjiiddehbgbgdagjdajgcdb 
#gicffjjcjbhjebjbacdaejjhjbhddeeiadcjead
fbfjbjeefdfb                            
----------------------------------------
-------------------------------------   
#a_/b                                   
#a_/f ,pi #c to a precision of #ejjj    
digits4                                 
4#ijfhiifhbaagajhibebigjciabhhbajgghffad
bjccabdjdfcgjbhgghdidbdfgfidjfaeijefcag 
#fidahdbjfbdidajfjcjjhddbhaaabjgbdjeccbf
bghecihjbfbdgjagbjdjgdhiiabfjaegcaecjic 
#ehjijgagjcejbchfhfhiigedfiajfhggjgafhcd
bjjffdghhcjacgbdfbjciihaiajcdhghahhjcbd 
#bedagadahjjaigddeicjdfafbbdihicjiabhdch
cdehecjbhifjihjecacjigficiaafffhheaaddg 
#jaghecgbbejibcbigeadgbjgabgjfibaehfcajh
icdffihfhafjecebdhcjdcfecbbfghbfhfahdgh 
#ddiijbfjdgiaheicbeajaiajcdigdiifidgciad
cecahdeeaacigdjjahagfjajechiifgecbceijc 
#dahdjciaacfjjdagicbhjijgjdhahbcafcacdeh
faiigafiefbjgbbggaibjjgjabfhbhchjcdciga 
#hghcdfgcdfgjfbhdidchacgdbgcdejgdgjecbif

jhfaahdhegbdgfefcggeaggbjjabeejdfcihecf 
#gcbcffgibicbghihdebgieiifgcbggigfeeeddi
bgagjjbjhccehcehefdhbcjagggggijbddicfff 
#gigdjggaegbaaeejhbiebcheagbhdidafgdhbhe
fcfgiaicbbheeihigfgidefefibiebdagbegjae 
#iddaehfbhhgjeaadhaeddgegieiecjgjiddadcj
icjhbdhbhjcchiiidejdjaagdbhbiadcfiefdhg 
#hffjbhediebgaiibibbhchgfbfciechhaddehdd
ddhehichbaiifbebgajcacfeaifbijbadaefbjd 
#gcgecafggidcdjcbgfgdciadghhahjgecabaiee
cichfbabcbfdbhagcihcfefhgfiaedgcjdjfcfc 
#jgficgcchdgehedeidcfdcbiihgdcadijfahfjb
icgedhfchdfdhiiebhaiehjbbjeejejcjdgeabi 
#caeheacdhgegdfgbdcfjabiaigdeheffeibgcea
jfijdfedahibchdjfdijaijhcabheijjdeiceah 
#hchgjggffifcfhbjgahgeaahcddcabfedciehcj
gcddhjhfjihfcejefbgfdgiddidiibfdadafhii 
#hhdajicbgihcfgjecfbijchijhchaabjdijbade
bfjjhgicgagffeebdjhjdifbigdhidaiiidjaei 
#jfdefjjdjeggbifcceedcaeejcbahfehaiicdid
dhhdbiicbbhiibcaifbifjacehgcfdecgbbahjb 
#cdgijecfdijdeebaicfaghjcbbfgiijdecdgied
iibfgfbebebehhdiebhchahifgefdcajjeieagh 
#dbaabbefgfddfbcjjdjgagdbggbiidjaeaedgac
ijbefjiibfidjbhjabdabbhjfhjgahjiaccggbc 

#jjhdiiaccfjfhfgihgbjfdeadgjjjccccdgddbd
bjbghdachgegjjbcjeadjfabcacbcdabicbbeeh 
#jgbcecdcicaaciheeaaeacfhbcahcjjiabdcgcj
gabgajjhhjbaibdeiicdibhcegibbdgfdhcicdh 
#hhggibcbcghihbejjgifccjcdagjbeefjihbcab
ccdbchfdaehjiabiifadfhdaebchachdddgchia 
#bhfjjcjchgbehicehjihaheejjaejabdhibgecc
cafffgfjjgedeagbjdejgaddaihffhhaegbbfaj 
#gbbhhcdjdhhfgacjbhgfeaiideccehgcdcchjgi
ecgcbadedhahhhaicffdhaidbdbdgbcdgjbddge 
#ighhffgfgfjagcdjddgafficdbhaeaigadcejdh
gecdjahedhgaighdihagaeghgighfbegdgjejee 
#ehfaahdhjigdeddabghcibdhhchjadifcdaibeb
cddbeahcjecichgcdebhfjbchjffidfjgejddhd 
#jhjfiidgefbdejebjjbjgfeecbcgibfjjegbhhi
eeejadffgbgcbhaaeegigdhidaigfbhbdhgahhd 
#chegiafdcfgbcdhbabhicigbgbhffedghigiaah
aaecibfgejfcjcbegbhajeaijdbhcbbhadccfgd 
#gcefdccbcgaibhjgicgchcafjhbcbbbjfdafeai
aijbieidcehagifebbdhafcfjjhibdhijajajgc 
#fffeaiffaejabijdbighgbdcbjhijddjcjabhgf
fbfaidddijddjafaegcghfjigabgcdiefggiehb 
#jcadhbacgjjaiafcgjedjchchegefhbfaeheffa
ajcegdgdcebejfhddhbfiggheabhhcgchddceeg 
#fgdebdjcdjgggbhhghiihgjbggbefijifdecbfb

fhigbhfbieiedbdbihgbcaaeeiddeedhdihhcei 
#jajiifbhaicegagifefbfabgfefhbfgdehcffgd
cggjdajebfbadijabbdiddejgajfehjcdcejcij 
#dcafhidjhibihdhfgicihbgbgdhacgjgibdejde
jdecebedhcaghjeedgjddfbdjghbcajiafhdgda 
#badeeahcighehgfahegdbhjchiigjhedaagchcg
bfbbbceccjajhfbieifegfbdcheaijgjeedbgia 
#jfbiifaciiegbffgefceegcjegbdajecbfgcfca
bjaacjcgdeahcfhhgccjeddccgahdiegdhiiija 
#ceeidhdgicbadfjhigdgfhfhacbiajdcjjcdcgf
gjccbdghgcicbjbgdcfchffddeehfjajacjihha 
#gfdfajacfhfbbfgfjbjcgjffggbjdfajfhggifd
ieacagbgghfajdfibjijbgadfechhhahbhjghad 
#baabacfgabbbgehbhdjbhidchgdehfafbgcfief
hgeadidfaacfffiihdacijbijjfahgcbjjjcjbc 
#fjcadifbaahegfgbceidgdfcgigheceibghgahe
afadeceefghhhjjbdiaaaajifeffjhigfjbhdbd 
#afcffbciceidfebbccajgeghhghchjahefgdcbf
iahfdhgdgcjcediiigbjgadfecabcfdcaefiece 
#eadigfhbfgidbdgefdfaabagiabagcajjdfigbd
afadaghajceefdfiggfijdaabcecieabdagcjdg 
#gcefagjfhdhdacjgbheecacaiifihfdifjcbjbe
febbgajahcfgjahcjdjgjcbicadffbaibghgebd 
#bejhaehebedhiegjgchgeedceijhdfdcfgghagd
ihaigghgaiijdggfcejhifhihdedjhafjijfjbg 

#dbhehhhafabbjbaahiagagejgfbffdgffiffbdh
afejjhcjejigabbbghbbjgddjhigacchajehgei 
#jcdfgegejjjbddaeafajabjgebgiejjbfhhjiab
bhicefijiffgiefgfjihdaihbfejajdejjfahaf 
#gabhjebaeicceccdjcejcjfddaacebbhafbiage
iegacbhbcdbcaahafcehbiffgeabbceahfhbaij 
#dbfjbcciadfihcfjdacagdhciibehfafcjiegde
bddijbjfjbfbfaffgfeahghhafjfagefabgfhaj 
#bhhffbdjdhebjbibaceijgiaheiigaieffjhehb
bhefadaigbegbcghgijfdjaghbfcfiaigfebiae 
#ciccjahhcghabidaibfcjgbaefcaeghjjdcbcif
afjdfjehabhhgidicdhibfdcjahajgfcdghfcai 
#ghiejhbjadficihhgfcfiddfdiejdjbeghbbfdh
jddfhddbhhbhhbchehaaaafeahcbjiihgdgjifa 
#eihbecifibjejbcgjaeegdecjjcbfgiffiiebgb
jfchhicfhhafjgeabcffbjibihageggbbgadhfh 
#ecfagaijchjaeheabafdedgaeffajbeiefiidca
hgcfcghhfchhjgehbacbibcecghdiijfggbeche 
#febaecieffhahaejfjiabfdjdeeaghghichfegj
geiiidddgcicihicjdbgjicibggiajecjijdicf 
#bbcegjebhhdajhdegahgegicjjdijbefdcehjcg
hecaigbcahdghggbaiacghgbgcgfhfjddceebjh 
#ijeigdbccebajgagehacbidcdaiddacfaffaffc
gabaafcgjchgedgfbbajcafdccifjahhgajdacc 
#jiaadbghhihgfcdbhehifcaahfcdhdejeaachaa

jefbfggifbfgcihifaacgibifcdcgachcgehbhh 
#jabdgjechdjahcgibbaaejcdhfcggiaahedhedd
agbhadgeaeideeffdbdfieahagcagjfhfbjhafc 
#fbagebeidiffiijghjiehjjecijdfjjhjbebecj
cadjbacgfebgdgadeggdjihcfhefhcdgebbecgc 
#igbbgigffehgicbiifghhigajjfadcdgaejceij
bjdgabcahghgiaihjihdibaacghjefdgbefiifg 
#bjifdhdiafiheddcdffjidfiggijbjcbijifigg
echbadafggcahcfejfbggffddgabeabdgadaeah 
#eifaehhcchechjebhfiijaacagffaiiefafafff
ajfchhchdiicafadbbdgedgjiehiagieechcifd 
#jbhfhjgcfgjcabbcgdabahieagebhfedifhhcjf
fdcchaabfhjcbbbgcehbhdbjfijabgdafcjajib 
#eafadhgdh                              
----------------------------------------
-------------------------------------   
,pi9999exp7#a7 to #bjjj digits4         
#bb4deiaeggahcfajdedgcdbgaebbjdedcgcejbg
ehicaeacciiffibbdibjcjjbeedjffibfjdjcii 
#aagiabcaheaigebgbgadcjcaecadejjgcadhhif
cgbgaffedafbgbgbjjjcfhdabdehghdhchbeghj 
#aigciiibgeafbgjiaaahebchfgacebidjhcdhib
afbccgfibdifgcjecfgeaffbeiifjaffhgbeedg 
#ggehhhjfjhgcgdbibjaahagaffaafacgbbdfaig
bjijddhifccacadeiibgcbgiadjiadhdjegfgfd 

#hhigecghdhhecddajbjjfdibeicdijcegeidgfc
dfiafebhfbidjjghdgciedjggeebihjaihbijbf 
#acabbdjbieaacgiiijfhhfebddbicaadfcceici
bhegafjgccbiedadiaecbecjacggbbgfgeeeahc 
#jfhgicjdcfbbhdbcaijhhbhfgigeghbigifgbfd
gahafdjjjgjdcegahajehcfdbfjfdadehjbabbc 
#ifijfigddgdihheafafaccahdjeajfbafcfdeei
efajhdaccjidbhiibhcabfcgeeghcfaehgdcjdf 
#ibiafdbdfjahccecaicdbccfdbjjcfabgbfchbd
ebcfbdgjcediegahcejdijgcdaifcjjbcedecdb 
#efdabijigibaaidcjcaafijhjjabibebbggjdid
djcfihhebfabhfaiaahcieaeifhgcaeiafihaja 
#iaaeacjgjbbagjdedgfdffagieibbebbdfhdeaj
ihcbjhgeifbaeiddbadhciihgdgddgbfdageicd 
#jbeficffhbfejceehjdgffdbcgdbcgjfagjheje
fddffdcbddegdifgchhhdaicfhhfhcaaeibebjj 
#eeefjbfdegjejeicbjaagjjicfbgfjeeceiehbf
fdabjaiihabjjdbgfacgjaibagdhfbjcaiajeai 
#ijeejfhafegadbehgjhaacdjebhhechjhfbiige
bgfjhefaddfjibfcadbdigfahbccfafhfhfidjf 
#eaebfffheifbdacbbiajfgifdgibjigibdedefa
ceidjjcbhcafbaehgbcecedhgcfddifacaighih 
#feacjgjbcjfbicegjcecgjhbhdfcfjeihdbcaeg
dhdidebchadfebbicdhdaicifeabbjicdjcbgff 
#fcabbabafhcihcgbiijcgfbfdgcedfdbhibcfeh

ihbjahhhgadbbfcjeahibbddfcgbhfihifhhieb 
#fbiicbbbajhbagjcbhdiiefdfdfehfhhdjihdjj
heddggibdfcffbcihjcdaffbfjicgiijfijbeif 
#hijjibgiafiidjfejcaahcfchgibjjbagigdibd
hfjcaicedabjdehiiigigjajffibieciabacbcc 
#eaffgbeiabahcbdhhhjfbjgfhiadfcgfbcdebih
becebeceadjdhihcbiigaccgdcfeagcejjjcifa 
#hhcabeddifhdiedbjcbhcjjeddbjabbcbjebcjg
gafihjffhdgjideigihgjijgheagghddeheffjf 
#dhcbijhdhdhjhfahbjgehgeeiaifjijcijejcje
hbgiacbghgafhgiahiaiadebbffiaaiejceedee 
#hfiggfggjaihbbfigdcaffbaiehjbbgaghgbfec
gbffabafgaededhcaeagcbibihehjchfhajdiie 
#djejdcebfddidehjchagadjccggfigdfahgcibh
diafiaaabbhacbdbdecigfeahjbjbjffcacbghh 
#hhacaeaeahcaiibiiifedbfaiagaeiecehcjjbh
ebgdgdfiihabhibhedhccegdjgigbaggbhgdcii 
#abjddaafgffcjjebcbfjfjjifdebacffdiffiee
gjbgfcehgfjg                            
----------------------------------------
-------------------------------------   
,pi#b to #ajjjj digits4                 
#i4hfifjddjajhicehfahhcddijiiihgfaeaacec
acfiidjgbdjgijfbfdaccdicgfbbjjddhbbdaib 
#jebdcjjaggcdjcgaheebbcahbdjbeiacggdjbca

ddjggggbcdhabbjcjjdfgbgfajfagfggiheaigf 
#fjiijciihefbjfegefcjegaejfjdabcbhdjcbhg
hjhficebgficdbafdicifffegaeaijddechgceb 
#faggidachbjbehbfjehaficdabeaeeibjdhcjih
ahhgcbgjjccjgfbfffgaajdcehiejhgaejdajjc 
#beghhecfeiebgfceggebhcgibbfhccahgdejhfd
jdedfcedabejbfigcgbieffiehccdbbghehaejj 
#jfcfebbgjiedgbdijheigefjgbffibfdgebggij
jebheccfdebbjfffihjhbfdaehifhggajegcbgh 
#hibiagdfijaedeeajjfibeddcbdegjcfddifefa
gbecgibhfjgfjfjjhadeigbehibbibcbdadbdjj 
#ddbieihacfahaddacgjfggggghaidgcifehcjca
gjheffcbghiegjgecdjgiiagadebcaehibfcgba 
#addfchbhbfddcbhebhjcgibhejcdhjiebcchiie
jcifhegdfjidhecdfjjijaggdbicbbjegiijcei 
#agceghbjdfegehjdaicafhfhbcjjbaifadfhiib
gjdbjfadbififcdffjjegiihdjceafdbacfedcj 
#diihdeccgbagcfeegbdjdfcfgfhdhhgfbfaeabb
iijbgjeiichjajbiiddfhhfahagafbfjihjacjh 
#gfecjjcgjfjaehcfiaihfgfbhfjjejgiicfdfhc
bbffigcaefhcfgageeehigaaihgebigebifcidi 
#afcaeeciddiaiedhchggjfhgbaacjghihffegee
ijihfecfcfefcjgfcfcjhhjfaaejjhcecgciahb 
#faajaghjhhgijbdiccggbieeghbafegidaebdgd
dbicchjdjghfggdjgeiagjiciaaigifgdcjagaj 

#aigjaaicciidghbjbbeeadgfcgjbbiafjhjbcef
cehfhiaefidiegihajceagbeaeaaehbhfchgbic 
#bggdibjjdgchdfbcjcbidafaiddiddiejdhbaab
bgdddfhcegbehacahefhfbeiafdidcecgfcibjj 
#bdejjcaedhbahgbjdahibdgfdcddcfdefccgcda
cfdhghdebideabgeiafcgbjjgdcjbfacfeahfhb 
#aagaccjdfjhiggghdhegbddcbefdbfcccjgbcfe
jdebcafgbcaefafjeaggcfibadhdihejijgabdd 
#hjhcfgbeajgifaihhdjbabdhibdeadjacifecad
gfcddeieggagfdgddhgeijfiedfcjdcgecagcec 
#jfgfhadcehdgeaefhhejidfhdbifeejejidacaa
iffjiigfaiicibheicebhggjfaffijfhjifiaeh 
#jbjeeceigghaheacbdjfffijhfaggjjebbagdhd
jiaceaajbdhdehhbfccjfecaijgddjcaihjbjjf 
#fhigbhfjigedahebdejafbcbjdfiijbidcjfdcg
afgaccabhbdfebgddadeeabhdhbaiggdgeaiddb 
#ebbicfgedhgddaececaahjjaahacicbedccdfch
cjggbbhfgjhdgehdbecghiffjaegbjacgjiifed 
#gfgijffbabhjagdjjigefiihefgjdddcgbejeeb
jgghjdifefcdaagjhadbijeadbidcgiefggdgdi 
#bdgebadidchbdbgbcbeaahjbhgfbgihjefbacic
bfcajiabghgbefafcecfjidbdghdebidghgaged 
#beafaebfeaiaccjbhcdjgdgcjbhbfcehgdbdccg
igchajhefhgfhfcidicjjiibeegbcgbjdijabbf 
#ihjgfiebhcigejbafhabhfbachegjhaaghbdagi

ajihhcdjacigeahacbjdfdaahhdeeeehdhaacjj 
#bjfdifhaibjajcjegcfihffddafbcahgjebaejb
eeafddhdejafdcddbehbbibaaejiiebbaabbfgf 
#hbcafiheidggjgjhhjgfjjbhcedbadieabgjbfj
ffghgihijhcagjebeafihjcgadigeejcbjgeedf 
#gcicecgijffdiiabgebhfbgjijdceacdcdeggaf
ejgdgcafhbajhafjaagfbhjdidhdeaegfeidfce 
#idagehhabfadebjdiechidefefebcfijadiijdj
cdaehacfidiedcdcgjiidfaaaadggjaadahjccg 
#jejhgejadbiiddfjaaccifidajiiifhdbhehcdg
gjcfdcdegehhcdibhfjabcabbejhgbbjbgcageg 
#eadaiiccdgccfbfehchhibbigdhdiehefaichhh
hcdbcieddaeaffgejbbahhdabcjibbhcdicaeca 
#ejcggedcccbchgbgfdjjdgcifgfgaijgiafebdh
hbdaaeabaijjceighaigfjecdbcfidagaebeijf 
#ceahicbfcbehaajcfegegaedihgcafifegijdbb
fgadighbahbabhdecaafajhjifjgfjjbcffdjaa 
#hbgjfjbidbijfdeihfhcggjfiajdibffeifggji
gbffjffcdjaedhgeghbjgjfchddfahghcjegebb 
#ggdgiheddabdhdhbfifiafjjjadbhhjcjaigaad
ghficeigcddghheiigdaefcihadbjafhdffhaha 
#hhbdechbhadbhdeeaebgaffefcdcchaedfacdhh
bfifdfabefcdjajehdbdbccahjfejdhhhaaidei 
#ihjfeagadfghdgjbjajggdeaagbggiddafbjbjc
aejcefcghebibgbehcggbfgjfbefbgjebhjjgbf 

#aafidfehfbhdedaaefgedibgibbdjgchbaghgdc
hiifciiddaeibihechcdibdcbdjbfdhbahgiddj 
#eigfjceceebijdfifghejdidhbjdgjfbdihdahi
ifcibaeahggecgahadfcbagibddaaahigcbcceb 
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jjcabbjfahagebjhhbjjicibadbcajhidfbaeab 
#baadbchjgdhjegebifbgfbagjiehjjfeiegbacf

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badfdijfjbgdebjcigfgejdjaceefcaideeebgh 

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dbcbagfbceeehhehbebgjdbfhihfcdaeabjcbeb 
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#hjhgffbcjgafahcgffhdedbhefebdcbffhjfbig
bfijfbfcghiggagfiecffgcbjebebgageichchc 

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egdfieiebcbdhfceecicfhgaibghdfegfbjhehf 
#feiageeaaeiddgaehjgfejhieigjeabagfebfjc

eejcjiijdhfhdedbdieegifijcghaciefefceid 
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#jjegfeefcbejaecgjjbbdfbfcgbfjagcghachah
eaibhccaacbbiddiehiggbedaagbgjebedfggae 

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ahidcdjbdjgbagjebihhbccehadeegbfadfebgf 
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ieafcbfbadfiicgiehcjjbahbjgdbgdhaigedhf 
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addjcigcgecibehedehfdgjagcafbccdechaeec 
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icgbehcfhgaaehfaiiachdbcedbjjecdjiagidj 
#jbijbifjhdbheadebacihiaiejhfdjcfdaiibed

ifcccfhfcbjcbdhjjhdbbhiaedgahjfdaiccjjf 
#hjibgaefcdceebhjbgcaccgjhegggfdhiigaaid
cfiihaiciaaefjdjcgaccfbjaeddhifhchfhhfe 
#hiicageidhfbhbfgg                      
----------------------------------------
-------------------------------------   
,the ,smallest ,pisot-,vijayaraghavan   
number4                                 
#a4cbdgagiegbddgdfjbeifjijhheddghjigcdjg
cddjdjefijagcccfdecdjaejej              
#cjbhbgheabdeedgeidjedfiicdgihaghgbhjcbi
iajibjiidgdbbjgdbeajhijbf               
#cijdehiggieeidcadgegjifgbcdgagedaffhcij
chhfgdahgeagcficaehdbeced               
#iijhbdffbbcedeccgbgcejdehihgiijiefhaejf
bggdeejihjbdhfbacjabafihi               
#daegebdegdedhfbejgefbfebdfajcfhichijdhc
iicbbfiiebjgdigeifbhbhhfh               
#eefijhaejgjdeacfifajiheccebeggbhaehfjcc
ddaadaibghbhbgcgfebifjcbi               
#icehdfgdbcajbhdhcbdafiebcijjfajhedccchb
ahejhcihajahjhiegcechgjdg               
#cicacdcifgcacgfgfdfjbajcafebgfhhicifcic
ecbeidciibdhcajcajiehciec               
#ggeaidbfjbhhggdjibgahfbjcchibhbjafaebee

ecbahbgjidgjfacjefgfabcih               
#hibjdfcgcjfegaifbigggafhhfcjhgfaeccbdhj
jaaagfhjgcaaffhdecbbggdca               
#effbhiifjgbffchcegbbajcfcdgjgjihchcgaei
hjbbccgajbacjihbdfhdijhfc               
#abificffcddcibddejjgaedaejdbijjjhaijcjf
gjehihdeccijecdfhhgbhgdjf               
#faieggefbfafgjfagfdbhhiaiciabcjhcgiahca
agafbbifjchhfadgfcehhjgcj               
#fcaejigdhcgfgehbdeibgjbhijacaiejieeaeef
jabbhjjcheiegfaedjaghdbae               
#agfahgddbaeieehfjiiffiibdgaadghjabjhbcg
ccfedacigcgaaiabibfdjegif               
#bdhdhcbcbbfcddbjjieibcjgcfcfajaeaejiacj
jcijjccbgaiaibjhefehddfbh               
----------------------------------------
-------------------------------------   
arctan7#a_/b7_/,pi1 to #ajbd digits4    
#j4adgehcfagfejdccbgdagedjajgfbbdgdjebei
eaacdebchhfiaghideiiibbcabh             
#fbgaadgfghfjbfccfgcagadbihidgghihjdjjgb
hagjbbehdabhbhaejbjeadefb               
#aebabegdicidiheghbbagdedadeiiidjahgiied
ageiajicbgidihgaeeiafjdjc               
#acdibgajiicdbjaedbefaeacdgcfhcbegebeaei

jhgaiadedijafhfcjjjdheade               
#jfeegdhejfiiadhceidaiifibjiighjbhciijid
figjcbjcggcgajdbafjghhaej               
#jeaeehjjijdgjfdcaacifaaigcbfahgbbdjhegi
agejjhiafabahcjjcjedejhjb               
#jgcfijdfciedcegebfjeehidhdbfbddcihjgahc
ggiiibgabefgacccehdegajcd               
#fgbegbbbcghjadbbificaegjdjhhfdfcjceijig
bbcfjihgcafaiadgfccjgichb               
#edfibcbebhdfghjdedbjeehefchifibdbfbcjdg
iiiccbeghibibifjchffbeijd               
#bdbibihhbcacdbdfffejbchddeabdicfihigaha
ecbbbhjfhghfeacaddfgbdgcf               
#bcehaebffiaejbjigdbcfgjjhebbefhahcbejjf
bbcddibfehchciffcbhfaiejf               
#ifbfbjdigdjjhbibahgeebhiiifgjhgibjibbja
fafjcjihcigcbahchadfjhich               
#ecdbbhhhaehggcefhgfcfcgbjeaicgighjidfge
fgjbeghdchfcgjhhfbebgifbf               
#aeaijahfhaefabafhebdgeachhcdiedbgjideea
ddiichgagjjhefibhifcdbfig               
#fcdgcjhaaggcbeaccjdefjebhfhfcghfehafihd
efijiddaibdahgififcgfiada               
#iedjgehbbedddcgdabecbcdjcgehagfagbbeajh
hbfichhfjccicbhhiighhjjae               

----------------------------------------
-------------------------------------   
product7#aa_/n9999#c1n#a44;infinity72   
this number is also                     
#a_/b                                   
cosh7#a_/b #c    ,pi7                   
-----------------                       
,pi to #bjjj digits it is444            
#b4dbhahigibjihhgjcbhgcfjdadcfagiadfceha
ahcfbiddghccijdigfcbgdiigdgbfdddgcdabjh 
#fhcfhabchjeejaegbjeijdchhachjfhjacgggje
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cdgijhfdaiiegibfbbbeiihdbejijijdjcghfhd 
#bfcdgjgfcjcjgfhchefcfhdbahefjhadbfgigfd
dfjeifjdcgifcjchaadbajdafbabgehddbibjeg 
#gjjafcjjgbhjiafihjajjgjceiffbahicdffabh
ihbcdcaibfgcgceehbicfebfifjbfhbebfiehbg 
#ifiddgebgigcebbghheefciihhdfcfbifaahfha
hiidiijghfbbbdhgihcefajicfcjdcgbeehaidh 
#fhaedjgabffhjhijciaajjdfbgbgceeacibggaf

jghccgdafcbjibhhaedbhhhiiidcigbjhdhghde 
#cbcdaebgjhjjdcjhjceedhffdjaegegfbjiaeei
ajhbjehbbffaddcadbeebagbfgjegjdbfcajjjg 
#bhjcbciidhhgjeacjfedjaeicgdagecgjjfbihc
haicabfaedfiigdjficagdhaciecedbchdchacj 
#cbefgchachdfbibaajiijbihhabigafcbbjdjbe
gegcgcjbffcgghgjbadjdghbfhjdffdfdffhcfb 
#bdfgegdgddaeedgeajccbdjddjaabigcbheibhi
agbjeghbfcfabjbdaeeiacdacbjfiadadbhfcee 
#jffbjgbcehciadjdjbidggihiafadeijiaidgdh
eaajfgdiahegebhihhajhdgbfjjdaiiafiagifj 
#bgehiceghhhgfbahdhggghjdbjjfffdadfdjjid
jfffffcfffeijfafieifbdhgdccbabidbacefei 
#jihhfbachbahjgbfhbebgegdcedjjgjddfibcgh
dafcggfahghiaibfgcfdeadhhcacbbgfgjbjgif 
#djciieagjeicefjejbfbafgbbahhhdgabfdcadf
bgecfhcdcbigghjdeieaaigejfabceeeefgceba 
#cgehgegdhjcegghggffeiggghafbcehaaffcjgi
aeeeefaachjibfahgeedeffbjcibbccgdfcccie 
#acfhgbhhjficfhdaahhdjeigdaeccgcebhcaaeb
cjdhbhhbdhdabhfbdhefghjajcfeefiiedhbfdi 
#eecfafcfdidichegjihfaaahgjgfedfhacjehda
baijaeagdfheffbaeigjighggcjiiajafcdehjh 
#jghdeeefeibeihjhiajjcbhhfcddcegdcgjihbe
jbfjecjiijbjbfdbefcfgajdghhdcffegieeiie 

#egjfaicdigadajdhhfffggdebfdfcbbibhdcecf
bgbbcbdeaiccdbhfgecaejjaaaehhdbhcjjeifc 
#edhjgdjgfbhjchehiadgbfagbgcgdcjjfciadbg
hbhfeedihibbafdgffdjfegffhhffbgejdaibea 
#hhdfgbcdheghfbaajedbibccjeceefejbjabieb
dbbjceiajggjggiaefbbcifjidceajfdceicibd 
#hfachchefibbjefhjdhidacfcjbgccacgedcbie
ieeafdbefiedegfgaabjjhddefjjejdibijbdha 
#dceideebiebiaeeadhcjdfaiaehdbifabedfhdi
daajaegbfgda                            
----------------------------------------
-------------------------------------   
exp7,pi99sqrt7#afc771 the ,ramanujan    
number1  to a precision of #bjjj digits4
                                        
#bfbecgdabfdjgfhgdc4iiiiiiiiiiiibejjgbei
gaihahefhhhgicechefccgccfiijhfbgjgecgda 
#jcghbajfdgiajaahfjgcabieaahacdfahfjfdej
daicjhchhgidigechfdjddijegbhgaddggaifha 
#dhebcbbdcbjciaafdghbiadhhfdbbhbgbjacaag
hcagjfejajdebbbfhghjadddhdaggjcdfifidfc 
#ceegjgfhagbchhgfhajjjibcgjfecieaichfejf
cfbgegfeghhheehbbcidhaadbgfiabajjhcjhhf 
#feaajgbhdgajfbcdfehaabihahcjabdeiacbhcf
ajjjfdihbffeibcfeagbfaghhcjhfcgajghfdeb 

#aieebhaedbgdffeajifaajjadgbejbjigijdfci
chagghgabegejjihjcfeggibbcjfdcabafeaaca 
#jhgchjeiibihbdbcceehdidefabciiefgfiiigh
dceifdhfdjifjjcbffdhbddcebacjfdiaeiicjc 
#bgjecjgecbefefhfahchhbfedhccjihjbhdffif
bdbhgchhdgeahdddcfhchecjgcdaaejdddfidgh 
#hdjjeidfddfiacafhbabjeibidfjededbafcged
hiahijjfjaejcefhgbhfbiccadjjfcfcbbfhadf 
#ceafabafcgfdhfdacadbicdbceafjjbadahjeac
ebhbhggcaifjagihaciaghhddjgaejffbiidiai 
#jicdifbggcifbjgbcdacecjbeegeghahjbhaahj
bajbjfcdjigdiicibchcgbijccjcfagcihaffcc 
#fjjcbbfabfbjhhfffdaagahjechcbheehigjjjb
gcegbbfdebccbhgjajfdiehfcfggbffihfhgchd 
#heiafefihbffbfagdaihheeaaefhddcjccbgcea
bcajcbdccjgegbgccafdiecfaebfbjdhbfhdgih 
#cjfjecihajjcaeggeihjbeaaaddeieggdahceif
dhijidbbjbjcdggaifgghdhcjhbbdejjgjaiaah 
#bjfajhdghggfbbegcehghehddjbcaijiaiecbaf
dbjgfcdadjjefhjciidcaedfebffgcgidcejbaf 
#iibacdgdggacbfaabheaiaccaghdiafjffehjfh
djcdhighghaddcacbbfgidajhcieaicfjbfejbh 
#ifjgbfecgbiabgfbbfichbdbgageeabghbgifec
gejgjjghdjjaaijjaibdagaccehcbgacdgjaeah 
#gefiebcahiejeggebbhifadifhbhbafejghbaff

heefjebahfbbbhcgfaeaajdebijgjdfeaihacej 
#fbdjfdjaefiieeejeefjggbcebgbcehihceibfg
iichbjijecbdahdjehiabgddhjadcidgdegjiej 
#fdgehfeeeaidgefjffcdgajgighcfffabibgfdg
ibjijifhgijcacahfeeedbhbgcbjfbfjfeicbdh 
#dacbfaebcgjehijjihbgecgjgaecgcfcjggbehj
habgeehbfibjhgbeiaehaijbjjejcigeaaibgbf 
#bhadbjeaebiehdhbhdfbhfjdhdjgadhjfgdiicc
gefhigedhafihigiaaffabejcbjgchciifcbidg 
#aigdgejffjhjgdciabbhbbeafajbihgaecabaec
ibhfgcbhijefdeeafheaajideajhejbdahfhhac 
#ceggecichcaiihhgeacafbegcddgiiidaajhaah
gdjjifggjfhbeggdejiejeibgieaggijjecdbbi 
#bbgfbeaceaegfgaciccebeecejhfihaicfdiech
aecchhbcihgjgeifgigfdgfhbejiacddbdbgbaa 
#ecgefbidfjicegbghjjbhjgdeaahhigcehddcab
beiidjgcehai                            
----------------------------------------
-------------------------------------   
,the ,robbins constant4  ref3 ,d4       
,robbins1 ,average distance between two 
points in a box1 ,amer4 ,mathematical   
,monthly1 #he4d #aigh p4#bgh            
#j4ffagjgahbbfgagfbceaeehcaaccbdhdacehag
dfdjjacegijiecfjdhjhiddbbid             

#giehdfdfacheigfcacjffebdhjgfhajgabjaeag
jiggecajgeidajigbdghfhjeh               
#afdcgbafhgdeccbdbjgbbihbdddbcbgfdjibbib
jfjghfjjjhfdhjeccbffichie               
#aebfidbjbhbaedbefibjhedjcdefajjcidfjfaf
chcddgbggaajgbfcibdjedfhi               
#gdcdeibcbbjfifieajdegagfghecjchgdhbchia
aaidhhgacjiaihajdgeeidbie               
#cabjedeehiaejcbfgecidjafdciccbjgijbiddg
cdgcdgijajacbijjaedeafffj               
#jfdbgcaddedfcaacfejciehefbebhifddcifdcg
cijjfbfejgceadcdgdiiaafec               
#cedjcgfcghfgegjeiehhbifiibgjjfcejjighch
fbhigdjdfbiaehdbgggcjfiee               
#gdcjahghhehjcgafdgjjageddfjaifgabacceih
bfbchgfeabjfeeeaejeiecchb               
#hbhaddbdibhaeicaefhjafdhafehabiiaaiabdf
hfhagdbechgifjfgaaddjhcch               
#eifbjcfbdeifhgeecbhgbjfihiiebgebjiadied
cgfhcaehgaihbfjgahcfeficb               
#giiahbaccgahefciddggeigiehfjjcadiecggcj
bcececgeiafhaigfdcbjhhffc               
#hgfabacgbcgdcdefeddeciafjdfffiabcfbhigb
efdedheedghiigdicfgidiijc               
#fghgdedaihjhgcjeijcijcigejdfdbihhjbdcgc

cihdcihabgjifebcbgbffchje               
#gggibigahgagcjiciafgaeabfcbeheghdecicgh
ibeifidggfaafgiegjbhedeca               
#begjheabjbabdfjiajagciahbdbbbfefgfeihch
fhjjgfjidiegghbfhgifebhed               
#d_/ajeag_/aje99b7a_/b7-b_/ce99c7a_/b7a_
/e99ln7#ab7a_/b77b_/e99ln7#bc7a_/b77-a_/
ae99,pi2                                
----------------------------------------
-------------------------------------   
,salem ,constant  ref 3 ,d4,h4 ,lehmer 1
 factorization of certain cyclotomic    
functions1 ,annals of ,math1 ,serie #b1 
vol #cd1 #aicc1 pp4 #dfa-dgi4  ,david   
,boyd1 ,small ,salem numbers1 ,duke     
,math ,journal1 vol #dd1 #aigg1 pp4     
#cae-cbh this number is an algebraic    
number of the #ajth degree4             
#a4agfbhjhahbeiiagejfeddjgjcchdgdjcejejf
icdaehjfefdfiebeihcjajfcdg              
#jbifhhcgfedhediifbjifhcjaaeehahaecidfei
bjgahacgicdgfhagfefbgadbi               
#icijdfijhjahidhjbebcafjjggeifegjedfjfbd
ahhgejdhifbcbeijgaggccdeg               
#aefgedhjifiigeeihabfggbhidjaabhgiaigbde

fihcgceaggfggdjbedgjahdjf               
#fbghfjcjjicaechccfifbfjggfbfhaiiaeigjdf
hcdfdffcbcbcajgabfefabdad               
#bbcjjhdgejihbgegecaghhaadidhcafheehfhec
ebdhcidcbdcdfejfidaadhihc               
#efjdheefgjiiiidaacabdhibdfeafdfaiiibhhi
dfejgjaeacigegjccabijdfbh               
#eifecafbcdjcfgcjhgjceicejfjchaahabjfaij
bjdcjjibdajheebchcihcjbad               
#iiechgbhgfaieiebjefgcigaehhfgejffaabicd
ehjgegegdcihjfeabdgjdgdab               
#bacdahhajfgihbiabeadhfccghjcgjabifhiafb
ebijdfeaieiaagfefegicideh               
#eadgedhfjhibdaffigdhiahafjgjbjdahhjjggi
ebgchbacjcbiagfcciijihahg               
#ddfdficaiaeedbbjigeifgehfahaagiadeeeeff
dbihcefdifeefeifchfjdejdc               
#dgaijfgbefdbfcbbiehjabbjhffdfffcdajbbdc
cjjdabcaajfcggecfijfaedhi               
#bhjdbfgjcjghbbbfcgcjbggjfhbehggadeghfgg
cfgdddecbijjcigeecgebacdd               
#cifieidebihbhjcjffgdcbfjhbjgcagahiidcde
cbdgebhibejehdaeigciddjdj               
#hbeddfaheafiacghjjffefcbcfihihaacgfffbi
edifgbgbghgdicfajafgfccfc               

----------------------------------------
-------------------------------------   
sin7#a7 to #ajbd digits4                
#j4hdadgjihdhjghifejffebejbcbafcjbihiiif
bbefcjfjgihcgajfefgbgeagjii             
#iaiajdjdciabciffhidhfcigdcedcjebfiehedc
dijcgijgibjfgdbicbeiaahib               
#jiiahihhhaaicdajcbggbibabdjidhjgiaieehb
fgfffjfiiiijggfdjaaighdjh               
#ghbgcbeffcdgdhdhjbhgjbihfefaegjagifbdee
cidhicegbibdfgjabgjhfdhfb               
#hajecchbjcjefacggbahbjchfhddiffggfafgdb
ffbcijacchbgeccigiefgfdbe               
#eefedggifcihigfdhbdcbhfijbgefifdbiabjfc
jjehcjcfeaebcjcabghbeebhi               
#hecbfdheacihaicdebaceigjieeifbjfbagbaad
hadddaghajegfjajgefgdacff               
#dhjeejjhiafgbffjehjdadjjghjfbcicjgjcgah
ggiefbfabhhhjdfcfjhagcdeb               
#dfefciadbjbebdjdahggfcdbjgdibjfiebjjgga
ccdghjihadbgijbadebfhbeef               
#fcbjhbccebaeddafjiafddbjijehibihgjbbdgc
chddfjddhigbcgacigiiabgdj               
#haibdgbejdhheedhgcaaicajcejfhaijhaeaecb
fjgdegcibiaaahccaifbhbaej               

#higcdhfhhaadbadebhchbbihfeabegjaffgchdj
gddeeaibcgefadcbbabijfjei               
#bdhbgcigjcfhahjaehefcjijedcbffghdfdcajg
ecabfchabagcbefgjaihefjaa               
#jfhcfjbhijahiejaidbaeafaffeeaiagiadeagb
jjdffhfeieigafiajgbaighje               
#hhedjfdfjjaiidjacgjadjecjiehjheebjebhje
becagaaccbcjedfafchcfcfja               
#hafiidgigaejjdheaejgicihchcjciefghafgid
hafabbadjbbjhiafihgajigdc               
----------------------------------------
-------------------------------------   
#b99997a_/d7 to #ajbd places4           
#a4ahibjgaaejjbgbajffgagdiiigjefjdgeiaeb
ibigbjibdfchagdacjaijjbbbdgaidffffhbbfi 
#agaeihgjghacddechacgfgcgafjcgcidggdgfib
acahfjfcgbfcfaghihdggefghecfjhfbechjagg 
#gejgjaeaeaadjceegjibbgcafbcdbhfhhhiibda
geddfjgaijhgajejchdiigbeeiajejjihcgajdd 
#ibjaedhdegcefgdehjijdhciidjicjijjjcdigg
ieijhjchdhifehhdcjjejdaaihgagjjicgijgih 
#bjihdfbebcecgcihabhagdjhahaacghjhbheebj
adhdbbajjfjiehicbdabddeicajcejegeaiaifc 
#jbidachcbfcdgdbhjbgihbddjhjbbhjjhbagbib
gbjehfaecfffcicgjdjjbchbjgcjhedefecjfgd 

#dggadheihhhgccdegfbgahfghchaafedgjdehgb
gfabgaaabfiihhfghdcdicjagehfadbdigjagjj 
#edacadeeadchiaiihgdcgffgfbagheafaghcagg
ihgcjgjdhbcfcahgcdgcdhdbahjecgaefihfhdb 
#fcfdhbgfajefbbhdggiiehfbhifccbicibhafhg
hgdgehfefjcdgcgiaiifdeidjjgefaedddcgaeg 
#dahijcjcihfigabidcjfbdhfbeceagcdabiaece
igecaabaeddfgdfaeijhfdggfjfeagddeiegjee 
#icjigiaaidfegefcihiagfhfigbagjbfbdigdge
cccfbiiahfjfecaaegjhcdicfhjgfihjdidhagj 
#fjgdcgfhdgdfgheehfebhbeejadahdfdigibdhi
jiieaefccghbiiheiejhgfdcecbciffbadgghif 
#edgiajdedahficdffahfacifadebahefciagjbf
cdafjdcedbbihefajhedicbfhgjhfhaeagagded 
#jdeededhecb                            
----------------------------------------
-------------------------------------   
sqrt7#c7_/b to #ejjj digits4            
4#hffjbedjcghddchfdfgfcgbcagjgebicfahcdg
adjbfbfijeaijcadjbgijcdhigbeiffejhdeddj 
#jjahedjegcjiccghfbdbhghcghacjgjgjggjcce
aeadihdigbedgdiidgfbcidjehbggefjdgahfhb 
#dbfdjdffaeieaaebgiajccihgdajjejedbccgdf
afcbejgfefagafccdeaffaddccbeccfabgccddf 
#jiahihefacebcefehcjahcicjgidjjiebdiicbf

hfhiibifidgcchbeagcgecbhhjbecghcjiagdjf 
#dhjcjejdgchjajiceieafbedaegbiadgfaigiia
fdihhidiabbedaddcegbcaiafdehfgcfabjhaii 
#bbicibgfihhccigijcaijiagfhccjeedbaehfhi
jddgahiaehjeajddaebgfbdejhcejaagfjceeeg 
#baddcdgiiedghahbhihedcehdbdijcfddigdffd
hbdbadaeajcicbjdcjaiidcficdhgfhgiaaehfe 
#hiaefigiifdiaejciaiceadchebfieffhdghafe
fjeahecfcbjjifbdeecchdaaeeiifddahghbjeg 
#jgaajjhcgacgfjeaahfdigacedaeebiidibbigc
giichcbaddddhhihjgciahigiaieaaddbgdbfde 
#aghhjafibfdjdjcbaijihfagbccjebihddhfadc
fdcbfcbjgfiaaccbcdibajjajeiggdbjggfcibb 
#jeijfdcbfgbeceageiehbejjhcddfdgbjggdjdb
cjcefchegaiiihhadfcdagcadghhgaiahjidgee 
#ejfcegdcaicgcdhhbgbiiabbehidbgedhgfhiej
fidjccbdhjieeihaaaadgheeebggfbadfahfaei 
#fjihhfiacabhjhaegcddbajafdbfhehcdadficb
dhjeiehebdhfidahaiggdgifijgbhghhceibffh 
#fhafefbieeddihbggabcabdagciceihhjbfagii
hhhdeifafagheaajaebfeadbjaibieggjgdheec 
#fbadggifjccecajabedgfjjhgihaeibicfchcag
iihgfdahcagaeedjjgeccbibfheecbcffdbfica 
#bifaacjbiaaajbeebjahdjaceadhgebciicfdjc
igcjhbijejbjhebfcdjigjjjiedghffgcajhgig 

#aidfhceabdffjbaacdeeagahdijfbcahfjjeeei
bfcjedbacddeeadihfjaeefjajejjcagecehhah 
#gbiabjbfjaibcgggeihfciiffhihhjgdecjecid
gdibggbaaaffcjjbjjidbefeahaeghjegbddcdb 
#cfdjgidfddjhafbbeicfcbecccbbfibdchgiieh
abhhcbadcfjeefbjdbajcdjjhchageheejjeadg 
#aeijceggegeiedhjhbabcjdeceaigjdjfdfjhde
agehgdfdhjfhbjjbjfihcebaeebjfbfhafafcea 
#edfabhhffcihjadfahhbihhgbgfhedggcdeeghg
ajgjbabaaecijiifafchjhgjjiecbabbefbbgdc 
#hgehdcacdhjebfffhdgajhajfhjbfigcjbabbhb
gjgjjfdbffejcijfhafgbdibhcfhbjcceaihhfg 
#aaadijeihjebadfbgfgbejhjjgjbigjbciggceg
gbfgbgdbjcfcehfhhbhacaahcbgdehccbjaafej 
#cjjffcbhgbjcejeciahdbibcdbbfaefehjbcchg
gbdjbejbjaabjcaiieeiheahaajicjadfjaaidc 
#cegeceeejhehdgjjadhdcdcgihcagejjbjddgfe
hajgaafgabfacighdagjccejfgcejibieajadah 
#jcehchajgchgagdfgbdghagigijdjdajfebbabi
cbcdgcdbfacjededacafgfejdcghcjfcjagcjcc 
#fjiggjbjbgiibjfddefdhgiigdjejjjchfjbhgb
jaaejbchcffbidjjjdegegahegddgcggbbdcieg 
#heiefdgcbiedaghedcfihjgeggfhhihhbjacajc
dahedbdjbcjcfdhdfiaceigibhddhgihgihacje 
#bjgiegfcbhghhhieciaafgdijbhcibjjaadejgf

fjbfjfidfgfhfhhggfhbhcbacebcdacdcgaddih 
#agbjfighgahccjcfhgbbbbgiaedcbchidffjfdi
bfeajgdjihfigfgjgcijhebehcjgdgfbgehhafd 
#eiifhdighbhgbbfacaieefbeiedfggjficdidfh
cdjhgaedfiaacbabcfhdfcajajcfdiedhciaeeg 
#gjfeigcbdbahhieggbiiegifaifdadchegcdgeg
dfcgjacbjdfjfhbbhbgjbjhbbbijgdeajjigbhg 
#dgjifebfcdejacjfihfcbcjdjecbecegaihjafj
cjchgeebigjichiiadbcifiicjigfbdihbjhbfj 
#fefihegfdfgiigajidhgjhcbcecgeicfaghidca
dgccjedbhjjhebaddcdhjbhiiaigjbfdecbjcga 
#edjeiaffidcchidjghacagicceghjjdaihchjab
bfgdfadigaifidccebihggaeggbadgagaedghfb 
#ibcedidajggcaeeefcjchcchgjccicbbheghijc
jcbcgbcgdihgeaggbgbbgifeffdcbgdeidibdff 
#hbhfcgchacadhgjgcfiaaghdcdegdahiaefhafh
jfdahacieaiabdbjjhaijcccehjcehiibdcfdbg 
#gibadfefgdfacjdffbjbhbighggfhbeefhcggcb
baiciagadafefgcccbbbggjijaiedajididiafd 
#gcacagbejhjheefajjhdfdhjigcjjhdffjcajea
feaihgbdccaaehbhjjaghccijijfgbaeejddhbh 
#dceccgadhbhbijhhgbbghfcdbejghaegigbjfbe
cfhjeiijcagcfddcbccabjffceifhcdgaccefdg 
#egbeaihgeidfdaefgdcjifdbjfeafbgeidbaeeh
dejgccegaahigahgcfdafgagbcbfbdajbahihjb 

#ihffdcecjhacgdeedgjgihdcacabfeabgidddhh
dbcebjhgeajjaegjacaadbcdeebdjbcdibjafba 
#hieccjccfhhgdhaceeabefceecjajcjdcaccadb
ahccjbejifddeddfiaiafjjjcjdefadcdeidjia 
#hdbaahjhdfieagfaefcdfeahdcjfhihdjegfcdg
cdihideifccagfdbaidifcejhbeiiabgeaabfdf 
#aghifhcadjhagbhdjeddefecgjefcciahijiigb
dhhfchbgbdadgdfjbaidjjfejhiaggcbdifjfgh 
#gjafcieghhgechicjjaadhbaagebacgiihdihha
cdeicfibjgdbcfigchfibeghaggfjdefajdhafe 
#giaieagbhceidgjgfdgaghebjcbfechjdbfdejh
gabdhhdjibgeaddbfchagbdgdghhgbdeeebaabe 
#abbfhfaeghjcahjijeagebdcehgiegijafciehi
iahcgeejccefhjejjccajhghcjbdghfhhcchhee 
#fdfbfdcfeabciifccfjjgidihbjidfhhhajjhje
gjeiaddidihdadggieabeehbhaiegaaahbdigec 
#adbejaecdhdjfdhfgecfbehfcieibiiaejeegfc
hggaadidbeidbfeehhagigcicjjaefaecahfcjd 
#iahdibeaedaeedbfjgbgighgdagigggheghfifg
hcafhdjdcbdajeciciefbcebeeaehaeghdgijhj 
#bagdcgddadhjgddgjggbibfggghjihgdiifdhdc
higgedifdbgfcgaagieabeijeafahhcdcjhjdgi 
#fbgfbgbaeiiggdecgajccihfieghecfcjgabbff
ddgebadjdhfgeeahihchfhccjjfgiehgfacacec 
#jfafaefbidgbecdhhagdjfehhhggbbhjcccjagh

haajbdaejcdebaabfidgdgdcchfdgfjffbheihh 
#dfbbifchceccjchbijhbgiadiccgebhbhfddgcb
idcjdeieadhbbaijfdbebhibihgigahgecbjabb 
#cjcajfjgbhjdbhjdbgbfbdfbefhdafdcggdbaga
icffhfjajjhdfaeehaebafcebhbdjcjjfhgicif 
#hdbhcfjgfgedcijhdbbjcijaafdfjcbcaaehaij
efdehhfadhhgfcfbgfeecgehjfiahdajfgcjbbh 
#hhiiibgcabhaajdjihjcaehahjjifjcafedhdbe
hajhaaehcjcdagcddcaeehcajgejffaibgadcfi 
#efebbdgefaeggfjhahbhajfdgeicgcggfedhhbh
ecaiiggeadidjbabhbcfgjbghhibeiefgdgffea 
#iiddcbaecegaehibebfjjciicababcdbdjfiadh
hagjdjajhhadhebhbbhfjfehajiiaecdadfhbca 
#dcdaaaaaciifaijhbjhagfehfbiiacffjfecihe
gjifcchadcdeghdjeijiaiihcbegeejaecejeda 
#cceaefdccbjjgieiaefceihjbjaidfaggjcdabi
egebbighhbdeiahdbgfcjgdjagcbeedfddidhbg 
#iihjiadhehbjdbdijfgheadfacfdhaajgjacahg
ebiggeiigcgejhjijebibgbcgjjihhbdhegifhh 
#cagaaidbhdhejiiaajgjbbgjeajdhbadedaidab
ddcjdicfjidchbdgdjjdagigbbdeifgbcidfeff 
#eedjchhhe                              
----------------------------------------
-------------------------------------   
sum7#a_/;binomial7#b99n1n71n#a44;infinit

y7 to #ajbd digits4                     
#j4gcfciihehgahgaejggijigieafhcfdibcdifj
fcabehcbijidigijefhbaiffebc             
#jhdgahahjbhjghfdjhahfidddahbdijbbeigdeh
bgbjcbahjadghcdfjagfijjee               
#dbbihfhdgggcbiddhiehhjfhjdaeiaeadbigicc
dcidafciihijigchjhcdbedjh               
#aebjjbiedfadfgbgffdigieedgeaegajefigbde
hheegdjieaiaaaihhfdhegihb               
#idcccbhehahcdhfadhgjdegijfdicbdfhjehbaa
igbidjgdagaafaihfgdfjafed               
#ddhedgihhiedcadbghfigdbibgbdibheihecbgd
gchjaeffeiacjeabedebcfgdi               
#daedeigggcddiajahfjdadddhigcgiccbbbbjhf
eejgcjdeheihjjejfeeaaaiah               
#icchjagccahijcbgjfhaheiegbicicggifcebef
ibibjbadadcfbhjeihafjhhgf               
#cjiafdgfefgfdjhibjjefcfhafijdagfebgibfe
baedjiafhbaigbejeebcbfddg               
#gfdfhacaeichcjdchjiihiehcijjjghgeefaacc
eciedijebadchebdacjcdfbae               
#cdegeiihedgijbaahjbdbahiheccdbeibgjchae
hdcfeghefgijaghhffcheaiji               
#iehihdgfdieghadfdeejdebabffdjgddcfhbeje
bdjhehgiceiiedebdbjbiaifh               

#fggdajjcaahaigdjchcejecefjfehdfdccfhgji
jbecgebiebdhehaddcfhjaejf               
#ffdbbdjjebaijceagdigehdcijeaefhbddbcdca
jgifegjfaafgbhfhciagecgjh               
#fdchagejcaiihccdecgiagaghfejaghgbiehcac
baffhjgdegebfbcfgheebheaj               
#afifibbcfjigegiebhbgfajccifhjggcbfjfigb
cjgcedcegcfafacfgebggjeih               
----------------------------------------
-------------------------------------   
sum7#a_/7n99binomial7#b99n1n7711n#a44;in
finity72 to #ajbd digits4               
#j4fjdeiighhjghjgbfafhfdfibgebedgchebddj
idfhhgdicfdbdfhehebcbidighd             
#fbgjggbgjdbaagifabbhjdaffbgcgcecchifahg
djhjdhbgjbbageaijbfecejhc               
#acddhjbgafeiidagcdchbajbjfbchgbgaddfijj
aeiabdeiihcfdfjgabeachaab               
#bbhjjddcaibbjjiadigdficcbabgcefehedehfh
hbhcfaadbghffgihbigbhfigd               
#dadiiibhgbgebbibbcjefhfheigcihgjbjhgcag
ieidaaaabefgdbihjaaijbdha               
#fgbhbaihcdcadgadahjdfadcijhgcibhiggiiab
ajgjbcdihhfiegfhhagheeabd               
#abcahifffjagcfebgijfbafgcdfjfhiihcccabi

hbfjiefhghigjjgeihbffghgh               
#djjgjbeiighcedijfjbbghiceidjijffidebhhe
cichjcbabaeddbjhigbdaccad               
#dfcgdgadheadfacchjjhdeebbecefbfdgiahhig
hbcaacgebcbiehgehbhdhifga               
#fdgjbaigcijgdefegadihdcgehejaahaccdagjj
cjicbceghbaegghfaieeaicbc               
#jahfciighbahecaggjcfcbhdghjjachhijeegbc
gfedhfgheahebfhbiieggghfd               
#ichdggadgdcfgbaifhbefghahiifaaafeebcgeg
hfabhhaijciicaghfcjdcgiec               
#jafaaejdfggbifajegebehjcdjihgfifejecjfc
echjfbidbhgbhgbafeebjbbei               
#iifccfjjghbheebgfbdfcgfeheggcebcffceadf
faidheeiagejicjbehgfcjefb               
#ifegbfbedgiigejahjfhgegfgigebfhjidcgdfi
hbejbaaahfbhiceeaghbibgfe               
#bedechcedadfcfibibdadaeejiebaaeihiajdeh
dfajcaecfjdbdbjeabiaeehih               
----------------------------------------
-------------------------------------   
sum7#a_/nn1n#a44;infinity72 to #ajbd    
places4                                 
#a4biabheiigjfbffcedjdjgbhbeijeiefjjedad
ihfaicfhbgdebbcagcajjjbdde              

#acfiddechgfebcddeeeehhagjdaabidbigjhihd
iiejgjibdhaedcjedhdajdhgd               
#aibhdhfdaigegiafceeeidgiacfifdifigdaefh
ghjbjgiigbiaggidhbgcjjijb               
#efdibcjeejgbjifffchabhdfgjabjecfhegdeig
hgjcjjabgghidabibhhbeceea               
#ggjbbbchccgecaicdegdibeiifgggifdhcjjhdi
ediaaajffifdigeejajeaigeg               
#dbiaafbajigjbaefaffiecbhigfhibdbghijjeh
jicijhadghhjidjcfgiicjeeh               
#iecebjjfccgafaajdfejidfchfjfhjhhfdiihfj
fecajbahecdabdgiaeigcgcje               
#bgajfhfhbdfebbdfggjccfhfjdfihgjbcdbjaif
ehcadcaccifhgchhagbiefhic               
#eecfheagihebadbjfffbfdafedchjfabbdefiid
jiffcefjdchhebciifichacjd               
#dhdjajaecbccheefihiedghiibbfadfeigjfhah
jgeccdbiabbhijiajjdiieacf               
#dajcehdgbcgdafgiffjiidjcgdbhhgbbhjijhbc
idgbdjcjabdbccgejfiffehgd               
#cadgfhcejbihcdgjjifeificjaihjgabbjeidae
dgdbciahhhdiedhhibjdcadgh               
#djcgchificeibhcbgddicgcjahfjahagegiebdf
haijiaceeifejfbjegfhdbgjj               
#hijgcbfedgacgbcchcdhdgahefbcbdhjddagcdb

cchefebgjeaacgddhbbjhfjfi               
#hchaafigjfddghifcaeedhjcaajhfhfhdfhjghj
gjajegjcdbcjjjjiedggffbhb               
#iibgjbbbfdbffahbbacjbiafjicddhejdibeefg
iiieababhagfejhajfbahjgcd               
----------------------------------------
-------------------------------------   
,the ,traveling ,salesman ,constant1    
conjectured to be is equal to           
#d_/aec997ab99sqrt7#b7799sqrt7#ea7 to   
#ajjj digits4                           
4#gadghbgjjgiabidbgbjahihdhgifbajhdjifgc
acdeeigjiddcjcaicifdegjjdaaedfaaggchcce 
#hgigjfggjbacdacjifbideccefaedgbbgeeegag
hiedcdabgdegjehfedahfghccbdebebaaddhdce 
#dbccgjafjgcdgdgdgbaefeejfaejbifcebbjbea
dfghheechgfcegegcfhdiddjadajdjbcbdbeeeb 
#cfdgjdffdhgijfajiiegjeaecichiehefcabbjh
dfcffigicdhgjhcaajaaffbjhddchaadhdghaff 
#ieccigbcejiigfjhbjbdhgafabfccedgbdfdgcd
ifeicahicfaebdidbgbbccabebejajghfagegbc 
#ijchejjidbhffahhefgggegcdgbjcjdcieicfjb
jjddafefbgjcdcfbhadcjgdcdfjabceaghgjdha 
#fjefehfeagajfhccifjiffehcbfbgefeebhbefd
ichjgiicjddcadijhgfhidgigjbbcjfbaaajccb 

#dbejgadgbiiadffgdjdhjahejjabhcecfafjbhd
jcaiagejffdhdidiaaeadjjedecjdidaigdabbg 
#fhbafadagaagicdcjaihacjaacgaabchbaajdci
ageijjhhhhdhghefbficdbfegdaaajgjhcdeedd 
#gcaiiiijdajhajajcfjgibifjeicidhicjcdggf
jchecchdjigfiabgfejecdfgaeaccieaeiebbif 
#dbejcdgccabbjgicccgddcgfjeiecabccagcegc
habfccjchfcighagffhjehacecfjabdbcbadbgg 
#jjgdjabiijcidehcdcjjcjdbcgfdfgefiacajhh
idacjheigbbedgdhbbjadcdbgcjfbbgffgdfbfj 
#bbdgbdhjaeffeiccjfgggedcedcfgeffddfbdef
aieaejaaehigjdjfhbhfdfedde              
----------------------------------------
-------------------------------------   
,the ,tribonacci constant1 is such that 
#a_/7a-x-x#b-x#c7 once expanded into a  
series will give coefficients           
proportional to approx4 c9999n and c    
7to #ajjj digits74                      
#a4hcibhfgeebadafaacbeeahebefdfecbhffjjd
bdaghgdfjigeibbdfgghgehfcidjdbjcbbbjhai 
#ffdbegchdcedaidbhcjgjadadaigihbfheibdji
gdafdaghdejgdfejgdcfidchcaedehbjdiieacg 
#ifbdifeeecifddfacfffabaedjbggigbfghaahi
dajdabaafjibbcbhbaeeiefjgahafgabahbcfei 

#hffebbgccghecghaefifihibebaagciegiadacb
bhgbajfahghihdjhebedieficaadecdiacdihec 
#deiegfagejceifebbacbchadbdgbgbgbbdagceh
ahggjjjfigijeeajbedijddifegajgdbebfedgg 
#bbhaajjfeihicgeeefcfcjicccjebhbfbcegech
eaigaiidbiiadecjjhbedffciggdgbijjehgjje 
#igddhaciaicafgbhbehdhhcifbfccbigjgjjfhg
bcfhcaabghcggejbejeegabbbgeaecbeieghidf 
#efjegjfhfdbbbhciahfeifihbidfiacefbcibbj
ddcaibdgfadgjfhhaadeagbfgffgabgdcifdadf 
#babegahdccdbffbcdjcijbahcebdideiajccbbg
bcajfaeacbhfiigjcjhjhjcfcjbbbccbdiigaje 
#bdcajgdgbcedbcaciigddchahbfefefjgceaidj
ceghgdiaagfbfhjebdecgjgibbaaajhdigajhjf 
#hgfdajjejaefedadgeffbbcejjhhheffeidigae
hbahcdahdhfhgadhjbijabeedcfiicdhjeacfgi 
#afejbehecjechghbgffffabfbbdcaggffcehbjj
idbiheejechgcbeiiafeaghggcjahddgbchhfjd 
#bfbbbcbdheghbjgibgbajdiafjahaghcgbefacb
jcdcihadcjbbgdecciigfbabca              
_/ #ai          #a_/ba_/c               
#d              ----  #a_/i #cc         
-----------------------  #a_/c          
#bg             _/        _/ #ai        
#a_/ba_/c                               

#i ----  #a_/i #cc                      
#bg             _/    in fact the n'th  
,tribonacci number is given by this     
,,exact formula4                        
----------------------------------------
-----------------------------  ,see 3   
http3_/_/www4labri4u-bordeaux4fr_/loeb_/
book_/#ibpl4html            ,comment    
calculer le nieme nombre de ,tribonacci 
,resume of a conference given in #aiic  
7,universite ,bordeaux ,i1 ,la,b,r,i74  
#a_/b #a_/c                 #a_/b #a_/c 
n              #a_/b #a_/c    7#a_/c    
7#ai  #c #cc   7     #a_/c 7#ai - #c #cc
   7     #a_/c7  7#ehf  #ajb #cc   7  #c
                                        
----------------------------------------
-----------------------------------     
#a_/b #b_/c                      #a_/b  
#a_/c                 7#ehf  #ajb #cc   
7     #d - #b 7#ehf  #ajb #cc   7   ,to 
get the actual n'th ,tribonacci number  
just round the result to the nearest    
integer4  ,here is the formula          
'lprinted'444                           

#c997a_/c997aic99cc7a_/b777a_/c7a_/c997a
i-c99cc7a_/b777a_/c7a_/c7n_/77#ehfajb99c
c7a                                     
_/#b777b_/c7d-b997ehfajb99cc7a_/b777a_/c
77997ehfajb99cc7a_/b777a_/c72  ,this    
formula has #b parts1 first the         
numerator is the root of 7x#c-x#b-x-#a7 
no surprise here1 but the denominator   
was obtained using ,,lll 7,pari-,gp7    
algorithm4 ,the thing is1 if you try to 
get a closed formula by doing the       
,z-transform or anything classical1 it  
won't work very well since the actual   
symbolic expression will be huge and    
won't simplify4  ,the numerical values  
of ,tribonacci numbers are c9999n       
essentially and the c here is one of the
 roots of 7x#c-x#b-x-#a71 then there is 
another constant c#b4 ,so the exact     
formula is c9999n_/c#b4  ,another way of
 doing 'exact formulas' are given by    
using   function the n'th term of the   
series expansion of #a_/7axx9999#b7 is  
#a - #b floor7#a_/c n  #b_/c7           
floor7#a_/c n  #a_/c7  floor7#a_/c n7   

----------------------------------------
-------------------------------------   
,the twin primes constant4              
#j4ffjafahaehdfhfiegcibghabaajjadeeegghd
cbfbc                                   
----------------------------------------
-------------------------------------   
,the ,varga constant1 also known to be  
the #a_/7one-ninth constant74           
#i4bhijbediaibjhahiahgeeddidceieagdejfaj
cag   ,one-ninth constant is            
#j4ajgfeciaibbfdhdegffaecbcddejijidgaije
hgigfejch                               
----------------------------------------
-------------------------------------   
#j4dgdidicgiihgibjfejccbejdfcfcbgihbifhe
eiedicgcbagbjbihbbhcccajbdh             
#fdeegibiagdhhchfjbgdbgefdabejejbadddahi
jcghdidbfbciedfdggebejdee               
#bjiigghebciejhhbghjhadhbaeibjhbefebjbia
baicjdaggjbhaieiihggihghg               
#fdjdcdbchjcecagiagjfbejafagjbebhjchdaee
cfhaigefgiahidhieibjhcheh   to #bef     
digits is also this closed expression4  
#b99997e_/d799sqrt7,pi799exp7,pi_/#h799;

,,gamma7#a_/d799997-b72                 
----------------------------------------
-------------------------------------   
-,zeta7#a1a_/b74  is also equal to      
-,zeta7#a_/b7997a_/b99;gamma#a_/b99ln7#h
99,pi7#a_/d99,pi74                      
#c4ibbfdfacibjiaeagbgdgaecaddfgadeiieacg
cjcbcigaejfejebjiefhbihdhe              
#bedgbjhjcaejcchbhdhhjfejebcajdadefiadjc
hjcdcgihhfgfdiifhdccbahef               
#jahgcgjgiffdhhffcbeecahggjjcjjbibggjhbh
dgibfgibfbicdcgigdjdgdgdc               
#defjfghcdibehgjiagfgddfbecjffhdedbahfjd
feddjibajgadicigjadjbjijh               
----------------------------------------
-------------------------------------   
-,zeta7-#a_/b7 to  #bef digits4         
#j4bjghhfbbdiggcedeffjagcjfgbecigjdicjbb
bfbfhecabhgfgbecgfajaaceega             
#jfadgbiaicbbibcdjdhgedcbffidjgccbaefdca
jiigefadabhfhiefeffacbfia               
#dfiddehcaaifegjefbcbidajiecajfafdjjaghj
gjjgjdacgejghcbjgeefffbdh               
#ghgghfibjffaejdfiadbhbiabcchcbeficgafac
fgggbichcfajideichghhhjij               

----------------------------------------
-------------------------------------   
,zeta7#b7 or ,pi9999#b_/f to #ajjjj     
places4                                 
#a4fddicdjffhdhbbfdcfdgbdaeafffdfjbeahib
ahidiijabjfgihdcggceeehbbicgjjjgdgjdjcb 
#jjhgchccfbhijjfaigehgjecjdjjdcahifbccga
ijfgifbhgbdfhgjjejjgghgiceajbidfccjhffb 
#gfhcagcccjicfggfbfjejiebeajjfhgbadjjedg
ifhaaeehgidhijcfjhbcbgggfaiaihdjgefdeeh 
#gfifcbcefcfgjigajjifidhijbjheicbjjhjeaf
cfdghhghcchhdfjddddeahdjeihbeadebejfhcc 
#hgfcadbbgfehgicibiehhjfcbjddgbaigijhdgg
cdjiajeijbjhcghbhiedibghbfchijcgigfcehc 
#cdcidbjdeaeiabjhahjiieicdedddhggdehgife
jjhhjhhidjhgjaaaafcdgajficafadfahdbhhgi 
#haedhfbddhceijiahcddhgegchgdbhcidjhbgfj
bhgefcbadcdfjajjacegffbjihbjdhgbjfijdjj 
#jgchbffcefjcjbdjbbhddfbifcjcbdeffjigaga
ieadbggbacaeieabeefgiihfaijhgaicaedciec 
#ebdajfchjddjgbadbaccifedgejehjaehgbcafe
hciidgfbdcdiadbbdccdhcfbijdhhgjjiffejei 
#hfbbfcjcdajieifgcfeebhaacgafgjcbfiaadih
ghdjcdcegafafjeggffgfcccjfgbebgcfhidbch 
#daffdjhhiecfbbgeiedjjggbgidgdhabgajbebj

dihcghdccbcjjagafegddhajcjbhfjdcdiffhhd 
#gidbafgbhdcceigbhaiiggichajjjhdffefjghj
ecgghbhheidgbghfbeicafahffdehhbibafjfeh 
#aicheibcbdaecbehjfdfaghabjahhdfdigggfbe
ihdiggefjfjichdfjfjeadfgfhehcdgbefbcaig 
#ajahcfcjadgihcgdhhifbaeibigjbgfcbcehgde
gchbbcjjfgiggiefgicaieaefeaiiffabchcfah 
#cffafhfeeffegigjjcgehegiciejchaicdfgjei
cicaadheidiaeiffcejehfbjhehebfchajfdedh 
#hgiehbjjjghigdcgagbaeficfegdijhbejhjceb
jdegdaacibhgfceecjidggjihfjhbcibbicihff 
#gjgejjebehjcfdecdjcaedabgcijgbgdbgbbhij
bbgdgigdbaegebabfebgbhffgijejdcefjhfddg 
#jaiebbagdcdhbifcjhjiedjgbjidjdchhhdecid
agdbjebghgaibficdaifbbhbjbdgdigeaeaahgd 
#acdgbghgeagiicffdgccfhgdhbjgebcceffjhhe
gicijgfeifaifjgijhabfeaaeiajejgbibaieeh 
#hddfacegbfdabebfadgeaeghjgafjiageaefhhe
cbgfhcbicffefedgfeehhabhdcfaaeaacdidhei 
#fgjjibbffbifigebbjfggghahajbiejjhgjbiad
jaebbeahcgdgdcacggbaggeecagijfgaiifgjja 
#aadiedgfhbibcfdbjgejbgjecdaafejdijeajgb
hfaahhhedgjggedegcegehedgdgjcbiegiaiijg 
#jhgaefabehabdjbeehhecjjjaifhihhgegbbdci
gagiechaaahjgicjgjhifdidcceieccefahcbge 

#gidfeaajcedffieffhbibhccjidejgdjfbjhdbe
cdfcjjhbgfjefhfahjbchagebchbcifeidfbdeh 
#bjgibjbdijfcgcghgbjhecjjdgicgiifgfjcefe
edcheaebacabjicfjehicdijdacjgediacaaiei 
#jdaicehggecahhhchjefgiabagacggbfdegjgbb
iiefceadbhabhajfehbafhcbjibhgbhfgdhcecg 
#hcjabhbedgcbiagjbhjbadcfhigfahjaifcgcfc
ahdihjeffhiiehfceecdajfhfdgdbeicjhjahhc 
#cfggdidfihffhchdbhidigggdjbgjecaagecehc
gehfjgdgdafidjegcgfcgaecebeafehgjahgaab 
#hjchfafdcbdfaghdhjabffgacibcfiaehedejdc
dddfdffdhdgaiejhgebhcjjfaiafbehchfgibeg 
#ghihibbihdddafebabedggaahagciahijegfbhf
jhdeghhfacfhdficcebichjjhbdgdaibidcbdci 
#cbcfbfdfhgfijhfgdibcaegfjidbjfaejbdihdj
jeficjbbhbdibcijfahcbdceahegiejaijcjjef 
#ageadehcegafegdccebbcbhbceafffadjdgfcia
bhcidjegfbfdgbdhhajjbjebjdahabjccghhfbf 
#bebcffeeeghhicggfcihaechbiadaeigfjcbcad
hjegjfeijfiaiddehcgjcadjbjeaechjehbaiba 
#iagbieijeeecighgidjjjghiidfaaihdfebgeda
dgjfhehecfejjeiifcffbfgeegjfaefiebedcag 
#gbecaeedchgfceagbgfbfaibdighhfafjifejgj
ddebdifcfigjbadjhhebfhdfhbfgicccgbggcce 
#cjdeajjdijdhddjiighjfbhddhejhbaajiidfah

bhggfgggbcceiadeifcfghdcigdajcacjejiehg 
#abiaccjijfhgdgdgaafaebffffijdhjjejcbheb
dfddhicbhijgihjiiiefibgccjbcffidgdddfif 
#ihjdjhigacegadjiaibaaiddbgbchifibdceeha
cghbgdcedbgbccejigcgcgbaifggejhadfhfegf 
#ffcddaiebddfdaagjjagiegeaiedfcbdjbgjjcc
hehcibgbigedhghgfcifbaaaggjicghdbaccded 
#cebhbddcajdgdbcebfabehbacbjdjabcjcfcabc
ihcbgccbdjbfedigefciaedjedjjddacfigiijf 
#gffbfgbeeibaejghbgigejhbdgjjghdcgdigcfd
iicihfiaigiehiecjbdchfifedjfhecafbbjeeh 
#hjffdgfgdgjigddehbgajaafjdchjhjhjjaifbc
egeegacehbbbahibfjfibbcgfgejcbbggefeceb 
#jfdjjhfagaghjaghjcehjgjhdhefgbegadggegb
ccceaabbjabjdagbehgcagjijedbebgbiegfaag 
#aeghefcfafgibcechhebcdcjfgeihdjhhejijci
ahaidabhcegbiaeheifjfheidaaeagcebhaeiai 
#hgjccbjihedjfhgeajehifehaibijegdfaigiif
jabafdagccaediibfghggjcjdbifiaiafjjaibd 
#dhdiiaeifdfdibdaibgeahdibgcbdihfegfacda
dabifafjjbdcebhcjaebeaedjccbjfihddbjccg 
#eeffafigfagceeccdcdeaechaabgfaghbcjcjda
eggchghfeagcegfjbjadehiijcjfiedfbjfijfe 
#ceagaacjjahhjgfjjjebjfejicijebaebhjjchi
jgfhdgjiidfifjahjhdabjcjiigdejdaahffgah 

#cbhfaahfedijajchefedeaecfafjjeihhchjjdd
ibdidebefeeghggfadjfdchighbffeggdbjfhcb 
#daaifgddigeacdffaeifbhgfehfidfbacfhffih
gaafdcfcabgefcecceadiebhfdfjjdhbhjegbgh 
#jjdaijdaihgjeajjhafeggjgbdjfeiacgehcihh
jjjebjcdcfcfbecdhjccdhihfijcheadiajcehe 
#cfhegjfcdehjjiehhbgabgjbhdicjjajiibhfhi
ceiihgadgdfhhdibchibfccdgcdhedbdedabfhh 
#gaeadfhfgeffafiagjfhaifgbjgebefiaccfjbb
jhijijbhgehjiebbiidefeidgfdaejfabdfjgch 
#beeiihhhebafgdefjeiifaaajcfjadjfcebhfab
fjdgaagdhjbccaeeebeffhaaeecibgdcgheeefg 
#hbagijeficfffeibiccgeaebieaidjicccbfjia
hdffcahedjfdifejjhceijfciahjehiagbcdhce 
#ddbbibiafbdcbdjccgejbfjbdbbfhgcddgeaihh
gjfcbaghgjcgcihceceeajcibdcfiaeeebeehge 
#acddegjeiiafddhccgifbddgahdgjghefaahihc
dggajbhjfdgbjjdcahifgchbjgiibjijdjcjhci 
#gffdefcbbfabfjecdacaheiffgfijggbaahddij
idbbdidjeffgdchbafjjfibgfagbceaecebejcj 
#jfdgheeaejbebdfiadjgfjdbigfgafdjefbhjhe
begedhbjgehecifijcfeaddabgbgjdcibifiidc 
#edccceedbghfcjedhdjjeiafhfhgchaihacejcg
bdjeeajbdacahjeagdegbhhgjbijbhjhhijjide 
#biaehcehechgjeghjiehahfdecdiicfcaaahigd

aecghahdichgfjcfbjdgbddbfgbacebeeifaigb 
#bjhiagfdeddcggeejaiabhhfgcffgbjheaehjjf
aeihjfhfigihehifhiighjiebdjaichicggffgc 
#ffiicijhdgfjehffagdjeecedagjifahejhgjig
eiaibfdcacjdbhcehieedchgiibbhbcccfbffij 
#jfedbjdddcfejhbfghfcdfgjdefichfdjeeaafe
jceggdhdccgijhfgcbidehdjjbbeichfegeibac 
#abibfjiijgbdgcdacejeaiebiaeeeeiifjcddhb
djefhgafhjigaaidhfdhiecadcjcehdfjbhcabb 
#fdbdfficjgfgfbfbfdfhgaaddcbbfebaagfgcbi
ifcghabjiddehcijbcfdihajdehcjiidceeaejd 
#dfcejhgffejabbfjdgiciddaigbaiagcbfbfcjb
dggahhdcddhhbgicdjcgebahceihhbhhcfadbce 
#adhicfiheifegcjhdjegffhcagaiedcaeggcgbi
iiaihccdeaihiccdicdejjcaejgibabidebaghg 
#cjicabeigicbdifgcibefaijaeebeigaiadhebd
jbhabediffjiihgdcdbaadddjdggdfjeecjdhgf 
#dgdhcbjcbcejaeeciedfhghceddhejebbjcbjec
geeeiffbieaifigceiaaejgjcgajjdbdgfajcgc 
#ggcgfehaficaagjabaeehdhhjhbecjbbhidibhd
hgbidaagdjedcbddfegaaechehgecaidgdcbdbe 
#bfadihchifcbibcdgagbfcicejbgcdiijacdbdg
jbficbeiijdigghigjdejhbbbjjfijehicbfcaf 
#hdhccfgjcfhhafgbgiffabhccafjfjfgchhbccj
aeddheeeigbbhcaeediecaebgijeabiddfcbidf 

#gcejfbieeaaiafgahdbbifihhdfeibfgahgajfa
iefihheiibbdibidhhgffcfjcfbgaagiabgcdia 
#bdbfchhfigbffejhgfhfdjhahhjfiijdiddffae
jiicichcjfaigaafdfgifaacjjaibahaifhcjbd 
#cfieijeaebegbbeccfbcgiibjeaacijhfbdjhjb
bcgbjbicgbeicadcjdbaagcghafdcijbebjjedb 
#jeecahacffcbeifegfgbcaiicdjejbiiheaddeg
ffgfdceafadddaeejjaeegbbajdjcahihhjeddj 
#fecgjibeejhcbjjhibcfebahhgbfdjhhcdfiggh
bdfcchcjajbajfecfgjhhcghgchejigjdjdjhaj 
#aahaidcbhficeebcjccgifidicjabeeafiahcdg
fbcajgjbhhheebcffgieghjfjcgjbbjeaggieje 
#jbhjjhfgjhhgaeifdabffhajefbegjgeffihhdf
ahfacihfaeibgddhgjajbgbbgjbhaacbhhjifbd 
#hfhajbicjdfeacehbjajifejchdbedfefceaiic
iddddecjifdggjicbehbdcgigadcdeheiabeejf 
#hggacjbijedffjbbgiafeahijfjdcdggdacaifb
idihhdjebjdjdiffggdabbdccafjifgddfbjfic 
#cbgbfddehabjdgdjceejfhagiaifaadfagehfed
jdeehggjcffchdigaeddjaieififhifaefbhifj 
#gccddegcebifabcaiidiajegefdciigbbejeafi
ghecghiijibeaacffadcadhgdcbdhggccfghidi 
#jaicebihhebbechjidijbebbeijcidgeaceaibc
gafddgdbdfaaebhfcdddhhhbjbjeaaeadcdhbeg 
#digffcjdciehdebiadbghjgfdjabhfdbbigcfce

jhchfaddihjbjfeciedbbjeeihicjecfecddbeh 
#djfbijbggibfacfdcjebbgjcahcjdefhbgijfji
ciigehgghfahdgeigfijbbdjegdfbjdaadfhdbj 
#ghijbhchidaefgihdaegjffhddhgegccbgcjacf
ddfdaebejcicfcjdibdbjgidegifijeihihbefc 
#fcdgidcgaghfjcjfbghjgegadbgedjedddfggaf
baaeajdcifdifaifaehiddeecdbjhghfbfefcig 
#cahggeacgbiijfbffbhidaghihhjidjigfjjejh
ceaeacbffheighigfiadcdejbcjdjebcehggidh 
#giegjcajhfaiggdeajeacdhbfahhjhbfffcdbbj
iaadcjjdcfcdghjacecgbfddbiceifjbdecbfaj 
#ihdgjfjifjiijgehgicidcjdafhiajhjaahcdch
cjfahidbbbfhcdcbjicbieifbhecicaiecaeeej 
#jgajcfghbhacfgeeeiafcbcifabfigdgdcijcif
ehjdddhjbgghhacjachadfcgidedaebhjaaggcg 
#fjfcicccagdgeefabcibehfeihbejiefbhhcaga
jagejdedcfidgbeajajchdegciahgbhehaidbic 
#efigcgadfjgicadbhabcabfieaaabdjhbeiicdb
gicccchbbcihchcbhgfajhddhhiiddaggbgejfa 
#bgfehiaifcihaddhfhdebfccjcgedhbjeehhdcc
hajccijaafcbdbigbibgjaihhhcffacciadjhba 
#hbhjhbdccaacgcbfgcdegijccjaahcdbfbhccic
ccagigejgajgfeffcfgaefhibccghddchgiiajf 
#gedgfhbdcihcccfgcbbfaeiajidjaecicdfiefb
aciecfeffefeggfdecdgihafjjiiedfiaicdcjf 

#cgcaabgefiddiahgcjabcibhbicjeighcfdcjfb
hieahfbjiihadfcdbbhbfcaeijcfcddhhcacjcb 
#hcahechgfghdjjbabdffadhadagcjihbbfdfigd
effegbheaihjeifahfdffbbbecefajehfejajfj 
#eeichicedbejihffbcdhffcffgbiahejjcifdhd
dbiaaeiibjedadcibefefadecbacjggfbgjadca 
#ajihfbehebfegdebfcdfjhdhbffagebjbidbajj
eiagahcagdghjgegjhbeiffaejfcjbcbfjicicb 
#dachcbiaeeeehafbaeedgajeefieddfeddaiddc
heaffbbeifeiijjdghhfaijffjgcjgcaegaajaf 
#ieheddgdgihghfichdcihcdjdjdaaaedfcbijgg
faijgidfdfgaaahfgcbdgafjcfbbaccdejjghhd 
#hbijdjfaafbgdcibgfcjbicicdaiiiehcgdcajh
acfecbifhidbfegahbeciecjhgcieadejaiahbj 
#icejefjficcegjdafcadcfaaeacdgfjdeiabfeg
fhefjjcbbifchdbdegihedihcagijgabfjiajaa 
#gigcahhcecadcadgjafibhbegbciddjjechcifh
fgecgjaedgbjicechghjhbcjbabfcbjefgiccge 
#ibfifdeehgffjefbfajgacbggggffaceebijcba
feafgjcfhjeihfaegbfadjefjfadhhdbgfbihbd 
#aibhbhbdfjjfjaaifjjgbbheddejdgbhfbbhhfe
baiiedgfbejgdcfcdigeahjeahjihfbceaediae 
#bbfhcdchfjeajecjjfaiiiffdjddhfcbcdfigaa
cabdcgcgfcgbdbdddidadibjddejfgicbgfagfg 
#fgiabehjfegfhffhfabcgifiafaghgbdcdigach

eijbjehbcegafjidbgebjagghdbgffgjiadfdfc 
#icaecieidahhdjagifhdccahfcjdfaebghdcahi
gchdjfbecbihffjadegefihhjjiafgfddhhaejh 
#baebhebeidcciihhfdheddgdehgahfibdcdjgec
ieccbhejbacifaehdghfdficcfggibgeghgadhi 
#ccdbidbfjedafibbhccgjggajfbajjbhifcefcf
daihhjeeahhgaegdicafbhgefhfbaagedbddfai 
#bbghaijcijbegdjiabfhhaeciaccgdacfghjgie
cfdijefgjfghfiejhhcacgbbfcegeiedceggedf 
#adaggadaedgaijbeebejehjghdfabccighidgai
biicbejjhehhbceaiheijjaaecbbihchjcgiafb 
#dbegiajjecefaaaddjcefccabefcbegfbajgcei
aijcfbfcegcbedbeehfbdajjacaehcigiebdged 
#cafiajfciicbegbicbgjdahjffebadeabgbface
jdijddfabddjgffbicjaahjdccjcidcaibgiccd 
#hajggaiafijbeaejaejcgaehdibfhhhbdhhehfa
icghieeehbhgaccifcejjibafhdfbhahfabdbgc 
#dgdjfjbeiciifjefhhcjeecbgcecahdgchafhjd
jbgjdgbhcccjbhjjdcadcdbiahegiccfibdjbgc 
#ceeggigjfagcaiejfjiiceeahaagbgfibgdcebj
afjdafhgfijbijdcdjdjbefdijhffhfjcgfbdgb 
#jdbijgbaeigjbeijdggbddijbcecafcjfcgjfjf
iiaichggjbddihifeigafgjcjcdegabdfiieiad 
#cidejabjgbcgdebfjdfaibgcjgdadhcdheeddjd
dgcbeiegfbaccigcabiigeddgfbgaddiejhbiei 

#cabeddeiebjebabbfbjghaigedbchdebgjafjei
aijfgcbhiehihgfdbdcajghcfbaicgbbdbcjbeh 
#iejdcdifbdccciedgggjdghdchajgihcggaabba
cidcbghibfjcefeaafhegaaibadcbhabgegcdac 
#djbcccbecbcgbbajdibedgiaahejagdhjcajbah
ihiedcjfadeaicajccahigcgbegjjjhijaebiij 
#jjdhchbfhjgadaijhfhiihaiiahajfgbgcffedb
dichhidggbjjedaccdgcgadheiaaifggcfeeeja 
#acdhgheicijeibaccghheefahjifbidadiieaii
jfaffcfefeaibfgcciebbbgjjbegdhbhjfddhaj 
#ihcbaieiiadchjddf                      
----------------------------------------
-------------------------------------   
,zeta7#c7 or ,apery constant to #bjjj   
places4                                 
#a4bjbjefijcaeieidbheciigchafaeaaddiiijg
fdihfbibcdjdihhhagibbgaeeecdahchbjeghfc 
#acjijahfdeehgcfjiccebehadfaiiaeggiebfjg
aidahdiaiieiihfgcbhcbacggfcifhcgbjgijja 
#fadecidaghbidicfjjffgaiaiaegeebbbdbdidb
dcifaefcijiffdajcbiaaeijieghjifeeadfeab 
#giiahdjeajegaebeeihhjaedcgajighaajbjcih
bgecbeffghgfjcebbccfihdidaffahaajegjadg 
#aegghfcidiigcgebcghebggicgjcjiefjbegjah
ecahbgijjjcjgfedgajgefcjdhhdccbjhfigaae 

#gcgdbchjgicddejcafjgfbecaggadeceddddaah
caaghahbbdigahebfcegjiahbddhiihgifbjcej 
#hccegefagbjbbfjccicghehgjcbhacabfghjgii
jjedaggcdhfiaaebecgjfefbcgjegddjiffbbag 
#abijbfbgcbjgcbcfadibbdbiacjdjebheeecgbc
dajccjggegggihjfdbdbjbdcjdhhbhaebajjjia 
#dfjbfechbbjfifbgaeebjbjhbbgdccejjajaebi
dhjaaihfijaagfbeieafgfcffiihagahceegebc 
#dhhjgjcgaieeegdbcdgbidjhceiebjhhfafffbj
begbhecgeehacjgibhbehfdhgbhbagcgjeeffai 
#fhihiebffbjahggfhajfbibjjhaggibcchacehg
fhbhdbfdabdcbdcadhjbhbagcfgdejfgbjficej 
#gfbfhiecjdcdeicicgejcbiffcfcggegejfbdgc
cbciibcdhbhhcajggccijebgfhjbjjgegihdcef 
#gicgaaejejijjejbgcffjdgaadjjheccejcdcfd
fgbbdhefecaeahaaggffahajibbbgiaiajbbdhh 
#cifhjjbfffjfefhgjeaijfbgeigchggcecegddd
dghggecgiafdadbgchacbbefiegcaifjbjahgdh 
#hdgdgajdfiiccfeffadjjhjficjcbefahecgahh
fjjgbgaheceidhbhhdghhfbdejdaheeedfdjheg 
#aeefcjjgabejijbgachfcdfhicgdafhbffedffe
ggbibfaaagahbdfjcfcjeffjdfecjjdgebbagjc 
#bfeacfciajehfihheghhdbdejdacdjjjgfagdgb
giacgahdbggdajhgejhfgijejahhifecifcefie 
#hfdcjhaifacgbiijbcbgdicdigjbdafbbfdedcc

ibcibibfgbghcfghfeegaeeehagggciffcggaia 
#bhadahbbdffdabfhffcdebhaajeeadjacafgcbe
cffhdahbgibiecgbffjejcdaeahebgjdhhjbhij 
#bfhcaehccdgieibjchgeeihdihhfaghfgjjeifc
gcajaegbgagbjjjaadccdgfgceaedahhbeebebd 
#ffcbfbigbjbechffadbeicgeiccdijaabdiedde
ahhhddehgihhcfecbcgfjejjfhfbafdbeibhdfa 
#hhjaacgafffffcejcefefjajjbeacabgebjjabd
cdfechaghhebbeaffdejefgcieejegchfcaebfc 
#gfeiedcjbhadfbbdbcjaggdgejaafgfhddegadi
fgjdhhjcddjbacjgcjbdabghgcaedjbijdbeaae 
#jiaiidjhghcdhfbjadbhjadjdjgafbaddfedghh
gdhaggehbfjdbjfffgcdjbejecbajggjbehcjah 
#chacbiichffigccaiidehdjfbcbijcifjegjcai
jibgbfdjfhchhjhefjhdjgdgchiefhccejebjid 
#aeadiagccjdhcfccjdggadcdehbeecibabbahbj
deafefjjdbgh                            
----------------------------------------
-------------------------------------   
,zeta7#d7 or ,pi9999#d_/ij to #ajjjj    
places4                                 
#a4jhbcbcbccgaaachaiaeafjjcfifedaafgijbg
gdgejieaiahgbfijgfhbigfbaedddabjfafahfi 
#fhhdfeefijifceidafiiiagbcbiijhacijhjdbg
dbdadehdjgaegdegjjdecdibhbjjceadgafbaib 

#jgjhgghcdhjiajhcgjbicbfahhgcdhbfagebgcf
jdbceejfbaicgcgejfagaaagdecdibifhfggejg 
#ccjgfjffhficdaahijehfbhccgiebgieabjccdd
iehijdfhhfbfbfidhbbjhcejcbihcfcbadijbje 
#cbabcieegbdhdffdfbbeejaaefffjdeehhbfhfg
hgfecejddiedceacgaigdieadhhfcacbhigdgbe 
#hhegeadeecbdgfahibcbdgdijhhcdcahcbafeei
ifbhiifdhjedjbjdihheeffjijfgajhacadedgb 
#dchbeaggebejdfiejbeebeadacbbjgfihjieeif
dggfhfbggebibigdjjcfbdfhhccfccecabbggeh 
#ffhjihccbccggdjbjhadihjgadbdajiegedegcb
cbgifhhbibbgfcbdidbbbeifgfhbadhgcafdbaj 
#acjjhcchcjdheghhhjbifjdihfggheibehceigg
babagacbeffeahhhjjbhajbgfchehabfiicachg 
#edeebcgghacgaccfcedfbfefhejeajbiihhibhg
jbcieafibjhfjgjccifafjhhbeiafiijaedfehh 
#hbiehidefjadhchdebedeicaaejgjagchbgiijb
jgaefiaiecdjieaehebgcehhcjaedbhgidccgic 
#gdfcijjhdjcdfaafdfcgbjdbfbgjbhbffhbdcfh
iibgibcjbdfafeaaaaddaajbfhfefjhiijfibed 
#bcdiafifchijfcebgeecjbiebfedijbideehdbj
bgfgfbegjbdaajahhfghjgdjheegidebjjebecg 
#faibccfiiigeaeibhjifbgiieggceghhaefjdbi
bdfbbgefdeifdchdhichgjgdaiiijbcgeijfgej 
#fjgcgaafafcbbadedgeghdejiiejcdfccjfebic

jeddfdbajjhdccjdaajdidbhhbcihgahbggidgb 
#bajgagiegafbjacigcdigejfbgccediaiahbgbf
ecgcaehjadbijgjfddaabbchihddfbhhhgdicdf 
#abcfbaghefhddbbjieccggfeffejfcabccicdah
cjiccaeijcffdhidcffgaceciejjebajhgiecaf 
#fbgifghgjabcbabfjfeaageccagchcbiaiidcea
idcjfcgjiddedehgjcjgehdahgeiffcaehhjdgj 
#igfbjigghccaiajeffbghcgcgdibbbfghfaiccf
icbbdcfbibabicgfhjbadggfbgfihhbbgbgeafa 
#afhaiefgdehacbfiibfgaeadgfgfaegdciiffde
cahefbhjgebbfcdgabgeffhcbjegcbfgfhgcjdc 
#fijhgbfeebfgdddiajgcjhhgdgadcdggagfhhac
bcheijebgdafhcjfacggcbjgdjhgfagecdeiifh 
#acafgaeeafajjbdgihihhbagiabbghdcbfjiidj
eebhdbdhiheceedhbbbheicijagbjffgafjbhde 
#bgjbdccdeegicciceegchacigdejhjhbeeggjbg
dhebediafcedibgeeajdibfedfefjbfiheefcae 
#cciidieghhiefjfgghcchahfejjadfifebbcbcj
eccecdffiiebcfiigfcdccgifgjdbbhgfdifddf 
#iefgedjdifebejciiggjfjghdcdfcaibieefchb
hgagcghfjjdgejdiggjbaefgfdbiadafcbjhcee 
#acdhjegibeffjfhbehjbiidfaigafaafffbfbif
bjfhgcihecfecccahbhggbceifefdejbeeeiacf 
#begbgfedcifihfiihhbfcbfbbebedfiebcfjahg
hgcghbcfhfgccedhbachgjjdbcegibhfaeiegjf 

#ebigfbfcdjgfaehdcaahhheeiicbeigiaicbgac
ebjjajbgdjbjgiahaagehgijgjgdgidafedgdhi 
#ajfahcfjgfagggcfghedgcdiaacjgbiafbhagfb
ijhhifcgbhggbjecfbagihecdhfhhjagdfiidgj 
#ajgchcgieicdbjaecechaibdccbicbfjaiiegdb
cgaadiddeeedacgfddfdihcchgdgccfjjiagbhf 
#icijedbgebehifdifibbacgcdchaeifhaeacebg
cfeaebcfhafjheicdgddicadcdhjgeijjcdebdh 
#jbbcgabdhcdjgjihicifdedaaechffadbaaiibg
ifehcjadcdeigjhjjeegefdjbaifjabacffchjf 
#ebgchijghgcfhdgaeehbdhabjhejccbdhccfghe
hjibggdfacdeifebdafigecjdefghaeciaachhh 
#ebahbaceidhfjihhbbhhdigaaigdhhdaiebhddf
ciajgediejffcecbebbfcigicifbefghjefhjbc 
#dabiidcfajbjhhbecjdfjcaecfifaejajfaeijg
fdjchhddijjbhgefdiaaaiefgfajjdhicdddghh 
#bfbffgfcfhiadhbhfhgihceedabhfjhfhjhchha
fggbehficfebijdjjgdigieeafgbacieijhhdcg 
#jadbgeaagfidabihedbgjjejggaidjjbggjdhee
cfhebgddbdgffjijehifcjdgceeecggbfigeceb 
#fjfhegdhcbgfgjdjfcidiaihdgdhcgfjaehdhed
ahcdegfbbfdcfhhfiajfhbeeahbhebfacjgbdhj 
#bhjaafdhcfcigeeiefafihfihcicbhiccdhegea
bahegeaedihjccejfijdfcgfcfajjideegbahgh 
#icedhdhdicddacgdgfagaidghdhiagjdijdfbji

ceeabidgifahcihffjhjiijbaijeibgcgfjafbj 
#afijdjhijhajfdfejhhihdfjchjcagedcgbaecd
jedafijfejabdddcagajeacagbaihfcjiagbcgh 
#ecgfjhidjfhcbcbfefjiidcgbhhehabfieijjde
fcacjcfbjejhdbdbffibcbejdgahdffeibhhaej 
#fhaccedhcahdcjggbfigjaiedbbcagdggibiacb
ddchfiaeidafbhbdhafdcgghjedehchcfhbhjah 
#abgiieaecejghaafjcgeihagddibhdhcfbfcjij
bjabbjiiggeffcggebiddbgbibcjcijejhdjehh 
#ebffeedghedbfcbdefaidfdicigibjcbahhfjhf
bgfadhieicahbjdabcgaeddbegbbfaijejfdjea 
#gdhhbidabafgjjfgcjdhibddbgghgabbdaihddi
ibdhjgiigiabdgcjigiidaieeaiejbibjjebhdc 
#egejcebeabifhjebcjihjbagdeehcdiejbeiiae
aijabjdgagjcgcfhicifiafbdhagdagbiifbigj 
#fgjjecaibddhjafccafdefddejaccccijgfijdh
ibehbefbjdjdijcadcfhiiheiajffgdeebcaddb 
#hhgcihadbdgdjbbaadfgffghjfddfegadgheggf
jgccbhjijjfjjgggcdhjhabfhdjjefjgaeahfag 
#cefffhajghbajcebcjchbbhjidghbjaeccihgfb
gjddhafiaaiadjjbfjegfaejjdaaegdaccdgegf 
#gdbhgifdaiaeehdhbighdfbgfdebejegeiahdhb
ijhjedjjddcgajifbhagiabdhhgbhdjfecbfdaa 
#jedfbgchdhgejbbfffeddhihegdjiadidbjabhj
gjibdhjhdccehiajfacdgbijagfeaacfdjbdeac 

#ddadbcijjijbhigcgbjhhejbidibicehdcfeibd
gedcjjjijhbgjicjhhfbahijajccfigdcbggccf 
#gfijfdehhibcjcdggcdbaedfeceedddjgeiddcd
cbihihadagbbihefcciedbhbjebeajafdebfbcc 
#iacgaiffaegjjheehbbdgbcicibeddijhhhhcjj
jiagfididagigicfgfhcdegiiihigddiabdafac 
#dficfbjgihbfcaffihhedbjihijhbjcjhbaabcd
dhhjaaiigihgebgjifjfbcaffeehhddghfghcde 
#efbijgffababaiiibcchcaahbibcejebibaaahi
ffdiabbgciiecaifgffhcgjadbgidhcajdgehig 
#eggaechcibbhjjceegfedejegffaciiicefbfbe
ijaegcecchddfgbjafbhghfjihcfgcddefeiefc 
#ijibhifhejbeaiajeibfbbgfgjaehcfaaffeeeg
ajfjdjfajcijjidfefdchcbjijcjiiahgigjejg 
#cgihhadebcbgbehhghahihccchcbjdebebdjjcg
afeijedchjgfciiechaachbhchhdecieegggehj 
#eaedidbaddibfgjffijhbabgdegacjigdbhcdhe
ihcjibiadjhhgbfdibaahdigdcbbffcgabbfhbd 
#hgeidhchifadihagafhefgebcghejafjfhdecbd
ceidjdggfaeadihiggiddgfcbhhjhgdijahidca 
#efgihjhgaihddfgeggcdefheabhbaaiajfhidcf
hfaicaigcdfgddbiggijjaiafhfajegjiifjgjd 
#bdgacifdjjbdiahgffhfafgaieefahehaiagdgc
cjiaghgecgfegcebabchefjgbihbbgajfdhjajf 
#ecgadigbbjicbeddcfbhcdbahdgdgdegfbhbagi

gbecaeebjjcijaajfghgfcagdjfgdidhfgdgfci 
#cdgihbhifgffahcccgegfhifbjbhfjdjajgaeeb
ihicgajafjjhcjijfjaijiibdabhjijhhfaddda 
#fdffedajehicejegfbdjgchgddjibcgieibgfgh
fggcadjedgjhihebjhebjfaadceecedjhjaffed 
#bbijfgbhedcjfhbeeddfigagfjgehabcajhjfjd
aefifgijcijagfaijhaabicfhgfefjgdhhifcda 
#ibjbidcgjgacbafgfggicchhfchgfaicjbhjjhe
dbejfjdfaijfbbbdgcfjdadeceafhgihegfjijc 
#baefdcjcdiifhcegffahaacaiiggdfecaeigahg
fiigagfajefdebgiebfebhhbjichhdaaeiiaegg 
#deafihediidbfbifghjiciicdgbgbfcedcjiici
aijcegidbfdegbjifhedcajgdcaaigdfhhjiihj 
#fbgdcgegjhfbaddbiegajibdfafegfhjbbgfhgd
jgegddbbadcbchjhgfgdhbejaeedgjfjegdbbha 
#cghbhgijbfdjhbbedbfcdiaijhjgihgjdiijggf
fhjacjjbjghhhghbciedjijafbffiegfbbbjbje 
#icgdbfgaceaffijaafgdhhhiahbggfcdhbbhjdj
edfcjjfdaafdghgdjddjfhdfjjjdajggibegfed 
#effjffaiicdabiicbgafhidfhcddefhaghhgggf
ibahbiddbgafidfecbdcbhgbdeadjcjfabgbhia 
#cjihfdcfejffihjdhichcejiiafbhfjbbdjifac
iajjggefhebfajijjihjchjafigefcbbedajhha 
#iejdeieajbbhjjbicccejdhijbcafdagcfejffb
hgjfiadcdajffdhidhgfabcgcfdfhfeefdibbfg 

#hchiddhggdaefbceedhaeiibedibcjebbfecfjd
dffhjbgfbjgdabdfdheahjcafcaehjbjheefgga 
#jbcahgcghbigcdjggfdfcefcegchccjhbageccc
fcjbehbhhgicaijaeeghbfdjjehdfhgdifebdeg 
#iifaejabiecjdahgehcjeafacgfcciahdgjhfgh
bhjheheggcejicgghefjecgiecdjfediheeadig 
#hffigbdabaadccicdjedgabccjiaefjiibcebbc
aehigfeiiedchfhfdfeedbhcaafhifjeeeccjbf 
#ehhhhddbdagcbjfbeedhijehajhcdejgfaiaige
aaahechcdicfbaafbchchhdeehfhdfcbdceehga 
#cbcaihcedgabjdbhcdadibheicigfejibcgifij
ajgiififjjbiagdjbcajgdijeebaiaaefjcdehj 
#jbceggbffcacghfaaahfbjbgdcijahdeeeghhdj
aicgfaibegdcfjdjhdgiaececgbfcecfafgabcc 
#gjegfebcabheeiifbahghifgggcibiaieghjada
ccgejbchbcbhehfjdaddibjehecibfachfiieaa 
#dehacghhhibeghfjfdeeacjehfccghjgbfhfbaf
afdiebjgajibiebbcaeabjjaihjhaihiahcjfea 
#cgahidjihebagbediiajaeicagfbidaagcafgeh
egfjicehcbhdihaejdicfbhhadgigdadcgigeej 
#bdefegdaaecijiihagccijacagfbaegidbbfaed
hahafabggdaeibfhediacbiiegebjajfahgffdb 
#hfbabagighjfghjihcfbhcaefhbacacfehfhcgg
cgdhbhhejabiifebjcgabifbjefgiehbeeiigji 
#fdfihacjdhibecjejcgbhabgaiahjhaeafgbacb

ffjedfgcjhacbjaebgagggeicfecigcefjdheaf 
#hbhhidggjebhdgahaggehjjijjfcchijcbbaehb
jaggibhbidgdjbiibcfebjaigghiicgchiadiba 
#aagdgjjcidijhhbgfajiedhaggbcbjeahddefbg
aejffecffhfbghcbddiiiijdjedjaggbiegbcij 
#gcbechfjgdcbahadhiiccjciheaebhidcffifdc
ggcebhigfccbdgdagafjhfcjidajijheieigcfc 
#aadghcahhcabbcigghfjiiibigbeageifccgghh
iheaabjhdfbccdiejhbhjhdcbhabjjdbhjehfjg 
#gahhaadadfbaigbifeejdgidgffajaighebifha
eifjdhddadgegbhhfiejhedjefgcafgfhaadcdh 
#hhiachifaebjcjjghhicdfdbcjibeieihgaebgc
jccghcbiaecddfegdfhidgcgddiecbjgieiicjc 
#fahbffcfhjcacbdbcjggdjechibddcfbcjhfbah
cghbbhabfdciiehifhdbdddehigbjjcfadeicfd 
#dfcfbcagbecbegbeifffaahchiejbgfchcecdbc
dddchifdcaadciaaaeccdchhfdeijbidaddiejj 
#gcbegagfcaeaafbfbhafdfdihbiigccfdfeahjb
iiggicahibhgiejjbbfhaajecfjcgbaidhciecc 
#jafagfdjchehdhdeedefdajhejgjfeaccabadhj
cdeajecbddchjiijdjcahaijieehdiaigdigjfa 
#ijdgdeebfcbhejibeeibhefjcbdgihihhejhdeb
jjhbedfeidehehjgeaifeegdjgiegeffdagbaea 
#dbibhcdegecjccddcchcbbdgccibjeeejcdjjha
icdbgadjghgeebcehhbcijchibhfhhbcajcaaaf 

#fheggdjgjijbcecciidjfbbjeecicececceigje
iiaehebidcifjfefcehcbbjebifdijbgdgaiied 
#iededefceegaajdhbgfjhcdbbicddiheajdcbjf
jajdbegfieifjagedheacjbigeajdfffideggbi 
#iaajjbggabbdfbehejgfdaibheabefdhfjcigfj
gggedejcagighfchbihjedcjfbededbdbaigaib 
#fejhdeediagiajbiegfbdeadjhbheefghijccbh
hdjcddcbejjajgecbeggaaijddaijcebdfgiajb 
#iaffceibgfecfhbeajcfehejdffbjhcgjeegeeh
bfjhcjejehdhdgeddddijidbjeeeagjbeeadjfa 
#bcfbieeagbagabfhegijcjjcggacabhceggfcef
ficaejbcgacdgighfffcciechehagdegdgajfdi 
#acfegiiecbcbdeedhgajgabdhdaibcidegajjee
bhadhabhaagbjbfefjdfhjfjdjifichahafacib 
#abjeiccajfjjhafagifaeejaeehcaahiheejbgj
idhdhbffibfbedgdjbehfbajbfdhhffjcbbddbc 
#dddbcighgaggcedidfafdfafagjiigcahddfide
ehgaabbihicggjachagdbgidjfdhjjcagafhdcf 
#hdidbagddhbjejcdcchadfgjdbhjaggbdciecde
gjcfgaaaiafgaagehegbhidcjfefjdafcaijedg 
#hcieccbfdddcaahabagdjfbhecjgbjaegcdddjf
jijdfciaefcadeagbdbfghcaibhcjhhcdaaibci 
#gcfbdhchcbibdgaaghbhbdgjfbdeahjfcijjagi
ibajcjbbhghbjaeachiifgcdgffbieceacgdfge 
#eedfgcfbbjjbcbcedbibjjghihcfaaecbjiifhd

abagdiahfbbeffejfcifebdjgfiabigdjajbedj 
#ghebfjeeifidhhecdfcgccbefcbghjcbbdiiibh
ahgfegeejgccjddbbbfegfdhjbihiaehcjdbigf 
#dgceaaiejeheheedgggjecegghdfadaiaaigihf
ahfhjccdiijijejciijeafhdbggddgafhdfbfde 
#hbgcjifacjehjchebdcbdggihddjchajfdhcbhd
djecebbfighcadcbgijfaddbcbhbbeddcdbfhfc 
#bfgfdfiffcacafacfaeigdcdbhfjdajgiejihgj
fgehbjfebijffdcbjeddfhabjghafejhgaihfjh 
#dajdebceaiffdiggaecdajjjbachifbbffciedb
bcghihffdbjcafheeifdfhaifddcbhhdhhdahde 
#bgjcfggheiaejgddidadehbfaagdbacahhcdgab
hiadefefehiaiegdibchbbhbahiijccffeibdjf 
#igbhajjbhceceffaihhheabhhjbiighheiiejha
igeibbbhgihjidchfdejhibfhgeeighecaghbfa 
#hedajbiaadbggcgjcddiiebjgjfbjhhadaacbgc
jjdbeebagcbdgiidjbfgfdfdiijaafbgiijaeei 
#bicfjbigdgeafjfcbejfeibbcgifhgiebiedhec
edbjaegadffdgfafhfbgihdbedggdedfciibcia 
#eehjgfhfaafjccibdfiifgfjabihhccchbhcjgi
echibbdcfdchfbbihccdfaahijgfbjfbjfgbjhg 
#jecejcfbdabighdbfjdbffafgghejigaiedecab
hcccfgjiigghiaahaiecbfjabbhgidijgibjifg 
#fadjibijddahddfhgfagbccdagbjifidifhhaaf
babjedejihdabidhigjigbebibdjccgbihccffg 

#jjfjdcifgcjhagbcfcjbjcidjfcjghebaddhhid
hbcdahdffhigigdcefjafhhjejdjgfcjjfcicfc 
#dagfbfgdchhfcehia                      
----------------------------------------
-------------------------------------   
,zeta7#e71 the                          
sum7#a_/n9999#e1n#a44;infinity7 to #eab 
digits4                                 
#a4jcfibggeeadccfiibfccacfedhfdegjcdafhj
egjhjiaiejaiabhaaigdaibfgg              
#ijchjcehighfbhadhdefjjdcajfeegaccccfcgi
fbjcdadffeeffjijdbhjjifag               
#giaeeigjhdahceaajgbahjjhgfddhffbhfccgah
jceceihcfcifbcfeabhhhhiha               
#ccebgfggebcihbgejcbjbbdcfhdegffdddffeie
haaeiiciagigggdejcibddfdc               
#iaiffffaeiffdjafbjecbebjejbaeaibbfgacea
befgheigdhfibhfjaigddgihd               
#cbjjfgbfhabigecjiaiijjggdfefeehfjaebfeg
cgcjjcgefaecbfhcadihigiga               
#icejcihcghehacaiibbhhdhhfdbecceajdbeafj
beajhdiijdcdfdjbidaagbdcb               
#gegfcdaejhafbccbbdefahfdiibgaddbgbbfdfa
daacjjgehjhfhcafiafdigiah               
----------------------------------------

-------------------------------------   
,zeta7#g7 to #eab places 3              
sum7#a_/n9999#g1n#a44;infinity72        
#a4jjhcdibggchaibbhbfhcigigedihdigifgeie
iihfcefjefebchgjfdagbhcacf              
#egafjadghcagceegcecdfjififhiacheacbcifh
ifadecfeadiajgdhhgbhfgggd               
#aihdjcceddjcaegihcjajccihdefbabajfidfce
hebdcijfehccecifdfgfiigef               
#gfiffiadbghjdcadcccidgdiebaecghijbhjjbe
ijdeeeaigicecajhcgjjhdbaj               
#gcbiciijdfajgjhefdabcefjehijfbbeiiggfji
hfidgedjgfcbjjjjdhafcbiea               
#behfgfibejfcjgcddacfcbeeefjacfjcjejjgcg
ccjbdacahgjcgieajbffbdggi               
#ciedfejbbedfgjdbjaeeajdjeehbbbdbcibejea
jhfhhcggbgjggdbfjjbaggajj               
#jaiedeegghihihcfjdfgdedjfabaiebfejgfedf
aafacefedhfgiaejjhjheheed               
----------------------------------------
-------------------------------------   
,zeta7#i7 or                            
sum7#a_/n9999#i1n#a44;infinity72        
#a4jjbjjhcibhbfjhbbaddaghebgfibcbdabjfjd
hefjeheacidhhhgefedheiffae              

#ijighejeccijbehcihiejcicjfiabgafiehfaeg
djhfjdgfehdgjfjbfadbecgci               
#gjgbbdcjaecjfiacbdihgfdbeajijibidhfhgfg
fedecifigidaedjghbfjbbifd               
#aeddhcfbejffhfbijefgjgcfdebafjaecadbddb
acbfccgeihhaeeehjebeiaded               
#jhdhijaeciebggdgdefaccdeajbhgdjfacbgdff
jfibgfccijjafbidbgjhfdbbj               
#aabcafbbjibdabfegeccbfbjedfbbicbaededff
eagiidejchffbgghbbcefdggf               
#affjccjbhadibcfdegjciicjaaaichcihejagaf
gibfjjbjfdibcjfigiehejide               
#hdegiffedhedjjbfideaahgeidhaefadcjcgegg
faedddccdccihciiheadaichc               
----------------------------------------
-------------------------------------   
,this number1 the ,product,cos,pi_/n1   
n1#c1;infinity  is the limit of an      
interesting figure in geometry43 ,if we 
take a circle1 inscribe a triangle1 then
 incribe another circle inside the      
triangle1 then inscribe a square inside 
the inner circle1 then inscribe another 
circle inside the square1 then inscribe 
a pentagon444  ,the radius of this      

figure 7the number of sides of the      
polygon increase with every             
step3triangle #c1 square #d1 pentagon   
#e1 4447 approaches a limit3            
,product,cos,pi_/n1 n1#c1;infinity ,is  
there any way to get an analytic        
solution to this8 ,like this would be   
the square root of ,pi or some          
combination of radicals and irrational  
numbers8 ,anyway1 ,thanks1 ,mounitra    
,chatterji mounitraseas4ucla4edu        
mentioned in december #aiie4 ,by        
,mounitra ,chatterji                    
4#aadidbjddhecbifbjjgjajdjaegdfieihgdbhc
jgieccgbjjhfceafhddjbccife2   maple     
routine --                              
product7cos7,pi_/n71n#c44;infinity72eval
f71#fd72                                
----------------------------------------
--------------------------  ,the request
 was sent by achim flammenkamp on ,tue  
,feb #bg #ji3je3ac ,,pst #aiif ,the     
email address is3                       
achimmathematik4uni-jena4de ,the number 
is #a4fjadjbbdcedihhgfaciccbe  7to #bd  

digits of precision74                   
-int7sqrt7x7_/log7#a-x71x#j44a72        
----------------------------------------
---------------------------             
4#bhcbfeabacajcjggcbehgfheedjdejhehhfhde
babcjgeiacdgidiefjicjcbddgfjbhibjgdffgjc
eeabjjgbhcdcbdfgahbff                   
#agbagidgjfcbfhgbchibcgdahbfebgcaifchiaa
fibiabahaigejhhhjfbdiebidbggbefaiagaidbd
bgcifgchdedeijhajffaf                   
#eabdgjbcbbeaceihdacchhibjbacchgececejfi
bcfbhffgjggehcgfachhehdhbbffibhccbgahhhb
ahfdgchiabebdgjfbfaic                   
#aacdafbjgedjcjjhjcghhadiifaebdjfehaejic
fffagabgedhgdebiabjgfibgijghhbfadfibejfi
cciaehhbdcggbejghjjjf                   
#hafiafhcfehdccechdhjecceahjdcadfdjejcjg
eddefbideggigeeehaggadbddghgbefbhbiaeg  
,there is a pattern in the binary       
expansion of this number4  ,the request 
was sent by ,b4,j4 ,mares on ,sun ,dec  
#c #ae3bj3ah ,,pst #aiie ,the email     
address is3 bjmaresteleport4com         
----------------------------------------
---------------  ,the request was sent  

by ,joe ,keane on ,sun ,sep #aj         
#je3jb3bf ,,pdt #aiie ,the email address
 is3 jgknetcom4com  ,the number to be   
tested is3                              
#a4chdcbifiafefghfiafcffjjjgjdfiahgbgeii
cfjbhidfgbbhjjcafhbhfchghjfijhhbajhjhcef
cde  ,the number of correct digits in   
the number3 #gi  ,the hints given by the
 user3  ,it's                           
log77#csqrt7#g77_/sqrt7#b77 or          
#a_/b99;arccosh7#h74                    
----------------------------------------
----------------  ,the request was sent 
by 7,mr47 ,b4,j4 ,mares on ,sat ,dec  #i
 #ai3aj3bg ,,pst #aiie ,the email       
address is3 bjmaresteleport4com  ,the   
number to be tested is3                 
4#hfbbdjabehfhjedegaeeggijbhcbdicideghef
egfdgdbgfhbiijideafjgabadeegcjfgdjeijeaf
dehjdbjchddadchfahac                    
#deabegbbijcjccjieheacijhaaadijijdcgbgje
fcaijdhcfgiieagccdfjiiceeffhfdbjcehaiaaa
iihgggbeifiebhhhcbdc  ,another binary   
pattern4                                
----------------------------------------

-----------------  ,the request was sent
 by ,jon ,borwein on ,sun ,nov  #e      
#jf3ji3bh ,,gmt #aiie ,the email address
 is3 jborweincecm4sfu4ca  ,the number to
 be tested is3                          
4#jaaahfhjjjcbhggajghgjjdfha  ,the      
number of correct digits in the number3 
#bj  ,the test7s7 to be performed on the
 number3 algebraic                      
----------------------------------------
----------------  #a4defgiajcajdfijg    
,the number of correct digits in the    
number3 #af  ,the test7s7 to be         
performed on the number3 algebraic      
gammamultiplicative gammaadditve        
zetamultiplicative zetaadditive         
psidigamma lineardependencesalvage  ,the
 hints given by the user3  p7#j7a q7#j7b
  p7i#a7sqrt7p7i799q7i77  i  #j1a1b144  
q7i#a77p7i7  q7i77_/#b  i  #j1a1b144  x 
lim   p7i7    lim   q7i7     i-inf      
i-inf                                   
----------------------------------------
----------------  ,the request was sent 
by ,olivier ,gerard on ,mon ,jan #bi    

#ah3dh3db ,,pst #aiif ,the email address
 is3 quadratureonco4techlink4fr  ,the   
number to be tested is3                 
#a4jfbeejhjedifbeeich  ,this number     
arises in the study of generalized ,zeta
 functions on non associative sets4     
----------------------------------------
----------------  ,the request was sent 
by ,michael ,mossinghoff  on ,fri ,feb  
#i #ad3dj3bh ,,pst #aiif ,the email     
address is3 mjmmath4appstate4edu  ,the  
number to be tested is3                 
#a4bifbajfeieiccji   7see below for     
#bejj digits of it74  ,as ,i mentioned  
in the original note1 it would be       
interesting to see if this number       
satisfies a simple polynomial of degree 
#cd4  ,the simplest polynomial ,i know  
of that it satisfies is                 
x#ch-x#cf-x#cd-x#bix#bh-x#bd-x#adx#aj-x#
i-x#d-x#ba  ,i found this during a      
search for polynomials with height #a1  
degree #ch1 and ,mahler measure  #a4c4  
,i also have a second new ,salem number 
that would be interesting to try4       

,thanks for running this6  ,best        
regards1  ,mike ,mossinghoff            
mjmmath4appstate4edu                    
#a4bifbajfeieiccjibafheaghcagiabecgedjdb
cjgbcgcfcibfagfhcfdfcdaigaedjjcegejgffc 
#eeecgbgjjdfjhajafbbeihdbbeeachifjhhejge
hhedgbachebccgebbifdgjceidhjcacjhbbbhfi 
#baccgggfaihedbjiihdjadfebgjccighfbhcadb
ehhebfbfecheheagfecdicbfbaiijijbdchdcbd 
#bihffhadcbfaffibgiaacieijhebfbgbicfgiaa
jdadeahigfbadhdfchaeiacdajhhjhejgdageeh 
#cgabbgdhjfjidbiaaaifgejiaijijjebeedbdfh
egbdbbbjabfgbijcebdegdgcghhcjceadfcbigh 
#ecaeiabaiefjidjbehjfbgegdbddjjgfcegbadi
ghdefieeabegdicdjgajhjfabgehjhbeebffbjd 
#ihhebfdjdgcbjhcjghbcgjdfehfeggjghjcgcch
dhfjhhchhahaehdihcbhaegdbebhigagghjhbfc 
#adgfibdhagcfghefhhcgjjbhiifhhigdaiiibfh
eegaehcfcdgddjbhfdefaiihjchbjihagehbhad 
#jajgcbbijecebfhidfgbaibhaadjjbebgddcefh
jbjceigijcacahecgggjbgbehifaaedceabfcjg 
#hdaeaigheagabdbaheiigfegigggcbfhiceggjc
eeehdjahdfhdeedeggbddgebbcgdigefhcciafj 
#ichbjeegeagehaaigfdadgdgabbieeaihjaabee
idiifeceiigjfhgbhjgjjdgedggcfheahbabgef 

#ibdgdihbjjfejdebjifjdfjfhhibeccceedhihi
fhaebcjbgdeceiibaihefggdhejfgeagjjcjjha 
#cdjdfadabcbidfjhhcfcfeijjahhghagegfhbhb
ghahcihcgfigbaaggffcfdihafhcejhafeedaef 
#ijdfjajbcadgghfhagbcfdjgbhihhcbghjicdae
iahfcdaaifbjbahdccjdghdffegahdbfaaddfdi 
#jdjgaeeacecffdhhdabhdghgjiifjaeeafabiji
fbfhacfcbhjjfiajfgefddjddededagijjajhhg 
#fgijhhajhgbhdhbbheiggibcghbaecdeghhdjhi
afbcjidhfchheacfcdhjicjhdcjbiaijfhgcgee 
#cechfeghggheefhdcceehadheddfejhjfcfcgih
ddeegcdfjiigajcjabbaddggaciabbbjffigfgf 
#aeacghgajjfcegbaeabejedgjdcfbdjfbaadjac
efchaijcgdfbcccfigjbgebdcefbehgggebhhfd 
#bgacbhifegccdhdbicffgbcfbaadjabfggaijhg
ageadfhbfafcgedjchgjfcffbafhggbgbfbhacb 
#bifahbcddcibhdeabeejfabgabcidedfiahbcfh
iahgffjcfbcafjfiahcgebbdififjcjahhdjbgg 
#gghijcbcgfihhbfahcaaadjjbfaeghfhbfjciie
eijefhijcijfieeefihdhcadjhddifdhbejcigb 
#ijfjaffahbgfedgcbhcbgeagbbgcgihbbiehcgg
abbgdciheichfhihcgjfagbbdieiiecibcbaicf 
#cdebheigahaghbafjjagjgbddibdaggfbdhbfei
gcggdbhdcehegeijfaebjbibdjjdffgdcfjgihc 
#eicdchgcbfbhdacaadbefbgfgfgjfcacicebeeb

jgfdhijhefjfaiiicbidbjfaaejcccfbaffcfbd 
#ffgbidbaaieiehcafaiaaagaaihcacdidejbeje
ddjijaaccjfhdbfhchjeafcgagcedcgbahjjbfg 
#fjgjejbedeigdgiicfcdgcjbhejheecahhbhgfe
bjjfjhabaafcabehgifdcjfegaghaahgiabcgbc 
#icihbfgjbhghcdcbjabcegdhiaeaffgdeiabahg
dicihgeefhbdacibhhhdhbidgdfjjgdhhbiigdc 
#ffchagafbaihdieaijaidfafajcfeiibedeiicb
dbjeadcdjchfccfihcbfebjicfbbijgaiechjcd 
#fafajchdfhfaiahgjfcfiaaddcaiaaiighhidhc
aaiffadbbbiedehfebdacifbjgehaijbejahdbc 
#djffbijhfdfajacaabieghbeceahdjicfhehgae
cjgfaggjbgdfaggacbfaejbjhffbjbcdfgfechd 
#ahiaiifhiccbagdgdeaahhjibhjbdggaihfjafa
cihcbghabjgejbacegbgcieffddbgeagbhgcjch 
#fhgijjfjhbdiagcffbadedidhcgafhigegjdiaa
ffhfjiggddcjiibeegbchbfeeiceaghgfjeggdb 
#beac                                   
----------------------------------------
---------------------  ,reference       
,philippe ,flajolet and ,andrew ,odlyzko
 in ,random ,mapping ,statistics you can
 have the article at                    
ftp3_/_/netlib4att4com_/netlib_/att_/mat
h_/odlyzko_/index4html                  

#a - ln7#a - #a_/;,e7   evalf71#ajbd72  
#a4dehfgeadechgjhahiajbafdcfdejfgcbigjah
gfiggijffibaidaddhcdiihafegibhadbjijggd 
#bjafabbjjddbhjieafiebedbjggbfebhihabadg
bbdiejdefejebagejhdhhbegaibcahggfijcigh 
#bhciehdgadedihafdiheedcibiejbfecgjeceig
cchebjcediceadhjbeedchbjihebifhgcbaiihf 
#cjbfjhjgfhbhiiacgeffdgjhiggjbhbbgcegdjg
aeejbjafhcijdffjhaddjccbibifacdjbhjiifb 
#ihggfafjjdbbjfgbdechfeebbjhhbibggdbfjib
edbjghdfbbehiibcejafdfhehhbhcgfbabadhhb 
#ghjahjcaeafefefhjhigcghgffbechdiehjhbcf
fdjddbbgajhgfhibghceegicajjiehffcabdcdg 
#fjhiabedidhhgiegcagggjgjgiicdcgcjgbbjff
hjafbjjefedehfiidefcffdedibhihgiaibgdhf 
#egeaehahhcacidfheghcdggfggbgcddjhfgifbf
ihdcfcgjebhdccjjcgfebgbechjbhggidfgfcdi 
#cgcghibeacafedidbdfjfcaibdgdeehfgafjfca
jhebjhadgghhiaeebhcbhbbbjcjagedfjhgdbic 
#jgbiehegidaifeafecfhajgbddgdcabdegfihgd
ibhacfcahgjcbbbahadcbhacbbcbcfihgfahfea 
#efjjceadhcdbhchccbahedeahabahcfagjfhjge
efbiedifgeeihiagiehcddihcafjeeeihafddcg 
#bjhcbeahichddffjciihbcjadgefagaiifagagi
abgjdjbgciceieabdjjdjfcgcfaifijdhcgbhjd 

#fhcdafcgafggbbicjgcbgjbjijcfegddhceicij
edbdhjcgaccegeicfbibjjaibibfcjfadffggag 
#ifhcaiedddf                            
#a  exp7-#a7                            
-----------                             
exp7-#a7 - #a                           
-#b4afciecdacgchfebhdhggjjjdjajbahjbcaag
jicgchfjbaejgibbgbeccegdabj             
----------------------------------------
-------------------------------------   
,the ,hard hexagons ,entropy ,constant  
,the hard-hexagons entropy constant is  
algebraic 7see below z number74  ,the   
value is 3                              
#a4ciedheigbdgicjbgcebbiejjffceffhhhjfhi
edajcgbhaddffaaijhagdgbaefacegfjhhjcehf 
#igggdfhihcghgcjhebgedbgijbffhifhefjgfhe
fegahdhdbbabdegaaieaafcicdihahbffidgjhc 
#bebedgagcgididgecdhfbbhabbiabfahgbhaeed
cdjabfafbgdgcefigcehegjihbcgefhabhihdad 
#idhhjjjaficdijcgbciiefebjidefhbecegbech
fccegbjjebaaibejgdgcihaajaeachjhfbhiffa 
#bfhacfghghcahhefcjdjdfhbgdgajgdggbjdfhf
hidgeffegehjijeecjbgjjfffgedjdifbdbggai 
#jfjhedecfadbbafhcfbificcjafijjccjicgbgf

ieffbabficihgbfhbcajdibcjdgddbhhbeadgha 
#gjbiffajgbgjjedbibhabbhjgiejfaccfcbaeej
iecehaagigdejgbccfiegdcdifcbeiicejgcddi 
#dijhidbdicbicjgedjhafbajeeecbhjfhfajfai
gjeedejcgieejggehjgbeecgfachehjccfaieje 
#bajiehifggbifiidaffcjidbfjafaeeffibebah
ediccfdgfifheeahbdbhahidfaejibheefdigdh 
#ejaceiijfibiaebegahccheajhjbabhaajdigee
ccihdgcffibgiadcihhibjdaheaceehcacjcege 
#fgcgajdfebbdhjgdedgddihbehchheahcbhgafg
cegadfjibjijgdcdjbheagdfegaefediijhbbib 
#iiihhdfabiifacgdgiiebcehccfejghfjggjeaf
jhghgifcabjbgchcejajbhieifehhajgfhbbddj 
#adfhabadgbfghijcehhhjjhheahaijecgdbhfff
fjeebggegbbgcdajecacbbeccg  ,taken from 
,the ,favorite mathematical constants of
 ,steven ,finch1 ,mathsoft ,inc4  ,the  
constant is given by this 7see z        
below7444                               
#abd   #a_/c                            
a 3 - --- #aa                           
#cfc                                    
#beja   #a_/b                           
b 3 ----- #cc                           
#aaigi           _/       #ca   #a_/c   

_/_/ #beja   #a_/b    #a_/c   _/ #beja  
#a_/b    #a_/ca_/c     c 3 #a_/d - ---  
#aa    ----- #cc     #a    - ----- #cc  
- #a                   #bdb       #aaigi
          _/      #aaigi          _/    
_/_/                                    
#a_/d   #g_/ab                          
#c    #aa  z#a 3 #c_/dd                 
----------------------------------------
--------------------------              
_/       #ca   #a_/c _/_/ #beja   #a_/b 
#a_/c   _/ #beja   #a_/b    #a_/cb_/c   
#a_/d - --- #aa    ----- #cc     #a    -
 ----- #cc    - #a                      
#bdb       #aaigi          _/           
#aaigi          _/   _/_/               
#a_/c #a_/b          #a_/c              
#a_/c     #b_/c #a_/b #a_/b #b     z#b 3
 7#a - 7#a - #a   7     7#b  #a     #b  
7#a  #a     #a   7   7   7              
#ca   #a_/c _/_/ #beja   #a_/b    #a_/c 
_/ #beja   #a_/b    #a_/c #a 3   #a_/d -
 --- #aa    ----- #cc     #a    - ----- 
#cc    - #a                  #bdb       
#aaigi          _/      #aaigi          

_/   _/                         #a_/c   
#a_/b          #a_/c            #a_/c   
#b_/c #a_/b #a_/b #b    z#c 3 7- #a -   
7#a - #a   7     7#b  #a     #b 7#a  #a 
#a   7   7   7                 #ca      
#a_/c _/_/ #beja   #a_/b    #a_/c   _/  
#beja   #a_/b    #a_/c #a 3   #a_/d -   
--- #aa    ----- #cc     #a    - -----  
#cc    - #a                  #bdb       
#aaigi          _/      #aaigi          
_/   _/                                 
#a_/c #a_/b       z#d 3 #a_/7a_/cc      
7#ajhi  #cgb #aa   7               _/   
#abd   #a_/c     _/    #abd   #a_/c     
#aecgf   #b_/ca_/ba_/b             #b - 
--- #aa     #b #a - --- #aa     ------  
#aa         7#a_/b                  #cfc
               #cfc         #acagfi     
_/   _/              #a_/d   #g_/ab  z 3
 #c_/dd #c    #aa                       
#a_/c #a_/b          #a_/c              
#a_/c     #b_/c #a_/b #a_/b #b      7#a 
- 7#a - #a   7     7#b  #a     #b 7#a   
#a     #a   7   7   7                   
#a_/c #a_/b          #a_/c              

#a_/c     #b_/c #a_/b #a_/b #b   _/     
7- #a - 7#a - #a   7     7#b  #a     #b 
7#a  #a     #a   7   7   7   _/  7      
_/         #b_/c                        
#a_/c #a_/b      #a    7#a_/cc 7#ajhi   
#cgb #aa   7          _/    #abd   #a_/c
     _/    #abd   #a_/c    #aecgf       
#b_/ca_/ba_/b        #b - --- #aa     #b
 #a - --- #aa     ------ #aa            
7#a_/b7             #cfc                
#cfc         #acagfi      _/   _/       
#ca   #a_/c _/_/ #beja   #a_/b    #a_/c 
_/ #beja   #a_/b    #a_/c #a 3   #a_/d -
 --- #aa    ----- #cc     #a    - ----- 
#cc    - #a                  #bdb       
#aaigi          _/      #aaigi          
_/   _/   evalf7z72                     
#a4ciedheigbdgicjbgcebbiejjffceffhhhjfhi
edajcgbhaddffaaijhagdgbafe              
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