,?is is a collection ( ma!matical       
con/ants444  ,^! numb]s h be5 d[nload$  
f3                                      
http3_/_/www4cecm4sfu4ca_/projects_/,,is
c_/,id4html  ,an index ( high precision 
tables ( functions c be f\nd at3        
http3_/_/www4cecm4sfu4ca_/projects_/,,is
c_/r9dex4html  ,y c f9d in=mation ab\t  
"s ( ! con/ants bel[ at3                
http3_/_/www4ma?s(4com_/asolve_/con/ant_
/con/ant4html  ,?ank y to ,simon ,pl\ffe
 7f ,simon ,fras] ,univ]sity7 = 8 k9d   
p]mission to di/ribute ?is collection ( 
con/ants4                               
----------------------------------------
-------------------------------------   
,cont5ts --------    #a-f_/7,pi#b7 to   
#ejjj digits4   #a_/log7#b7 ! inv]se ( !
 natural log>i?m ( #b to #bjjj places4  
#a_/sqrt7#b99,pi7 to #ajbd digits4      
sum7#a_/b7bn71n#j44;inf9ity74 to #ajbd  
digits4   #c_/7,pi99,pi7 to #bjjj       
digits4   arctan7#a_/b7 to #ajjj digits4
   ,! ,art9's ,con/ant                  
product7#a-a_/7p9999#b-p71pprime7   ,!  

,backh\se con/ant   ,! ,b]/e9 ,con/ant  
,! ,catalan ,con/ant   ,! ,*amp]n[ne    
,con/ant   ,copel&-,erdos con/ant       
cos7#a7 to #aejjj digits4   ,! cube root
 ( #c to #bjjj places4   #b99997a_/c7 to
 #bjjj places   ,zeta7#a1b7 ot !        
d]ivative ( ,zeta function at #b4   ,!  
,dubois-,raymond con/ant   exp7#a_/;e7  
to #bjjj places4   ,gomp]tz 7#ahbe7     
con/ant   exp7#b7 to #ejjj digits4      
exp7,e7 to #bjjj places4                
exp7-#a79999;exp7-#a7 to #bjjj digits4  
,! 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    
,feig5baum r$uction p>amet]   ,feig5baum
 bifurcation velocity con/ant           
,frans5-,rob9son con/ant4   gamma or    
,eul] con/ant   ,,gamma7#a_/c7 to #bef  
digits4   ,,gamma7#a_/d7 to #eab digits4

   ,! ,eul] con/ant squ>$ to #bjjj      
digits4   ,,gamma7#b_/c7 to #bef places 
gamma cub$4 to #ajbd digits4            
,,gamma7#c_/d7 to #bef places4          
gamma99997exp7#a7 to #ajbd digits4      
#b9999sqrt7#b7 a transc5d5tal numb] to  
#bjjj digits4   ,si7,pi7 or ! ,gibbs    
,con/ant to #ajbd places4   ,!          
,gauss-,kuzm9-,wirs+ con/ant4   ,! gold5
 ratio 3 7#asqrt7#e77_/b to #bjjjj      
places4   ,! ,golomb con/ant4           
,gro!ndieck's majorant4   #a_/,w7#a71 ! 
inv]se ( ! omega numb] 3 ,w7#a74        
,kh9*9 con/ant to #ajbd digits4         
,l&au-,ramanujan con/ant   ,! ,lehm]    
con/ant to #ajjj digits4   ,lemniscate  
con/ant or ,gauss con/ant4   ,! ,l5gyel 
con/ant4   ,! ,levy con/ant4   log7#aj7 
! natural log>i?m ( #aj to #bjjj digits4
   ,! log#aj ( #b to #bjjj digits4      
log7#b71 natural log>i?m ( #b to #bjjj  
places4   log7#b7 squ>$ to #bjjj digits4
   log7#b99,pi7 to #bjjj places4        
log7#c71 natural log>i?m ( #c to #bjjj  
places4   log7#d7_/log7#c7 to #ajbd     

places4   -log7gamma7 to #ajbd digits4  
,! log ( ! log ( #b to #bjjj digits1    
absolute value4   log7,pi7 natural      
log>i?m ( ,pi to #bjjj places4   ,!     
,madelung con/ant   ,m9imal y (         
,,gamma7x7                              
,bessel,i7#a1b7_/;,bessel,i7#j1b72   ,! 
omega con/ant or ,w7#a74   #a_/7"o-n9?  
con/ant7   ,! ,p>k+ or ,r5yi con/ant4   
,pi_/#b99sqrt7#c7 to #bjjj digits4      
,pi9999exp7#a7 to #bjjj digits4   ,pi#b 
to #ajjjj digits4   ,! ,smalle/         
,pisot-,vijay>aghavan numb]4            
arctan7#a_/b7_/,pi1 to #ajbd digits4    
product7#aa_/n9999#c1n#a44;inf9ity7     
exp7,pi99sqrt7#afc771 ! ,ramanujan numb]
   ,! ,robb9s con/ant   ,salem ,con/ant 
s97#a7 to #ajbd digits4   #b99997a_/d7  
to #ajbd places4   sqrt7#c7_/b to #ejjj 
digits4                                 
sum7#a_/;b9omial7#b99n1n71n#a44;inf9ity7
 to #ajbd digits4                       
sum7#a_/7n99b9omial7#b99n1n7711n#a44;inf
9ity72 to #ajbd digits4                 
sum7#a_/nn1n#a44;inf9ity72 to #ajbd     

places4   ,! ,travel+ ,salesman ,con/ant
   ,! ,tribonacci con/ant   ,! tw9      
primes con/ant4   ,! ,v>ga con/ant1 !   
"o_/n9? con/ant   -,zeta7#a1a_/b74      
-,zeta7-#a_/b7 to  #bef digits4         
,zeta7#b7 or ,pi9999#b_/f to #ajjjj     
places4   ,zeta7#c7 or ,ap]y con/ant to 
#bjjj places4   ,zeta7#d7 or            
,pi9999#d_/ij to #ajjjj places4         
,zeta7#e71 !                            
sum7#a_/n9999#e1n#a44;inf9ity7 to #eab  
digits4   ,zeta7#g7 to #eab places 3    
sum7#a_/n9999#g1n#a44;inf9ity7          
,zeta7#i7 or                            
sum7#a_/n9999#i1n#a44;inf9ity7   ,! ,h>d
 hexagons ,entropy ,con/ant             
----------------------------------------
-------------------------------------   
#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 ! inv]se ( ! natural log>i?m
 ( #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;inf9ity74 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
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effhgghbiefhebdcdbjicigceiecigfbdgfbeia 
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cjiafcbfifjfehcciehedjijgcdifeejjdedfci 
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eiahfbbhcifecfiaeejehbejdghffcfjbgahcfj 
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ageabhgdjdagcdaggicadaieaddbcdcggdeecgh 
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gagbcjijhcdhdijibfhbdfdecfaddjaebhachbg 
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#ceigebeffjhdcihehjahcgiehhdbibfdheagceg
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cjhgjcdfheccgigceeiejdbfdegdahccaagghgj 
#deaebhggajjhefeaeijeggecfbdcedjbgcicdgb
cijchgajdcfe                            

----------------------------------------
-------------------------------------   
arctan7#a_/b7 to #ajjj digits4          
#j4dfcfdgfjijjjhjfaafbadbefbcadfabaddjbj
bhecgjedbhfabjbfchajiccjhhgbjaighfdafegd
                                        
#agjecjjfjjbhcihdhhghibeefebihebbeaaijhc
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igcbjgheeeeegbjhfjadfcbbgefebdbfgcjebah 
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gghecghjabhbcfccabagafdeghadfhcfijjiicc 
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fafbjhddaeicgbcfabgcaffhgfihafhgjbgeica 
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aefjighcfchjjiecgdiicbjidhfjgccidiahfdc 
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eggjfbbgeddafbhejbbbgcgbcgaaigddgccffig 
#gcaheajfidjachaabfiiegggibefbgdhbeffjji
fbaafgbfgdhaaebgbhbgbbebjgbbeigbfhdbaeg 
#ajaiehggefbjiagjaeeggfhgjihffedbffhijcd

diceahjedgbhijjecgjghchabdbabhedgidcjcj 
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bbfigggaaeahibggagbbfddfcd              
----------------------------------------
-------------------------------------   
,! ,art9's ,con/ant4                    
product7#a-a_/7p9999#b-p71pprime7       
,ref]5ce 3 ,wr5*1 ,john ,w41 ,jr4       
,evaluation ( ,art9's con/ant & !       
tw9-prime con/ant4 7,engli%7 ,ma?4      
,comp4 #ae #aifa #cif--cih4             
#j4cgcieehacfaibjbbhhjedgbhjedcdfeafdaea
aafbibdi                                
----------------------------------------
-------------------------------------   
,! ,backh\se con/ant calculat$ by       
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#acjj places4                           
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ecaghcgdhdgacaedjbgjgjbd                
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jiibeadcfaifcdgacedhfjgg                
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diihhjbedhfibcjgjehjhebh                
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ge2                                     
----------------------------------------
-------------------------------------   
,! ,b]/e9 ,con/ant4                     
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dbdhbacihabcf                           
----------------------------------------
-------------------------------------   
,! ,catalan ,con/ant4  ,z calculat$ by  
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type ?is4   catalan 3                   
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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                             
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dhgfjeaedjf                             
#hbjbbfbeeiejdhabdhehidaahgcehcdfhbbijdb
cjhcfaeeedg                             
#fidggggjhcaidjhgdhabdiafgdhibijjfeicfif
afdaffbcdcf                             
#hcgjgedcifchchideaddjaaieefdhgchacdbiba
bbihbjjacjb                             
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cebdigiihej                             
#eiahffigbbjabcafdhigajdhcjgjagicebhaabb
bhiffceeabh                             
#cagdcgcebcicjaadjbgibchihjhgddefifdhcji
jacbjghggfe                             
#hghecfbcjacedbhjjjafbijeehggbiejjfgiehg
faghbdgcgdh                             
#gacghjfjjdbbjhddecdfjdejfdgjbddcbehjhea
fdgggagcijc                             
#aifjbhfeeechcbhbhadaeiaebdhgcebfccjgaej

eacadghhbhd                             
#diiibchffcbdcaihajfccfeaebdccaacbadfcij
jicccfbaeia                             
#fjgdddhbibcdegaggdedhagafiehjahafhhijja
gebhefdejdf                             
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achehfgegfe                             
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debfggjefca                             
#fjfjigcgfhdfedjhcfidbcjbjchafhbfcibdehe
giajgdjdhgj                             
#hgihgghebdbfadjhfhgaeaghegehjajjfjbcfha
gjcdiagiggc                             
#cfbbaiffbiecggahiachecaafgciifeefehheia
bafdfbhjaee                             
#hbfbihgcedacgfccfjgfjgcjbjjdeeiabjbidff
egcdgegaheb                             
#gdecaafcchdgggfdhfhchbdhejdaafcjafjebbg
jhfiddddbgj                             
#cfddbeabdbcfcigahadiiibcdifjhchieiafhbe
hjcfafdgdih                             
#hajdbfcidhchijjdbidjeejdcaejbaicabfhfdb
cjjeiiibibf                             
#cfaedjfdibfbffdahfehceidijdbdicgaebcfbb
jfhdjcidjcg                             

#jajhfhjgdjjihddjjjaeabdfecdcecejfgbcchd
edfidfcegfj                             
#baahfgfbaadcdadbdgfaaghcdajdcabgcjfaafg
hbbdhhccifi                             
#iaeecijiacajigcbcajffghaaagdheecgfgijbg
bcahdejgfed                             
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ahhaabeiadf                             
#bjcigdbcdjaabehhbacaefijgeagegedigifehb
bifhihdfbca                             
#cbibegbgcagecchcjbcaceccbcbhgjjefeiefhh
ecdagebjdeg                             
#cicbgehahceacihbcdgfghjjibfadbfebajgdga
jdefffhgfca                             
#cdcbeffgbgeibihiaiebedhhdijcghjijdfedfd
hhbfhegebjd                             
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hfcdjijadaf                             
#hbhedhdebdbdhjicajdcdcbaggbdghhgghbdhgc
idhfjfahjjb                             
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effadibhiec                             
#ajhfgbdhfjhjdhdbjdcgffcgicfbcjbacfdehag
gihjbbgbjhh                             
#bgchjgagcfgaabiihbbbhijfiabegfcjbgggiaf

bfeajcegfbe                             
#ggjchajdbhhfhjcgfjedfcfcjcccgidjcfgcggf
ifgddgegaga                             
#iahgabhjciedcgjiffdachggbbffbfhhihcgcaa
aaafjbjjdea                             
#heicigcagdgfdfbaedbhchdfjafbaddebfeecgb
jbibeebjeae                             
#jdidahbhjjcjcbeejbfgegijchbebghfacifcce
gbgbjfejhij                             
#cfghbjagfbehegcfcffjbdeifddiadeccebhada
jcgbeafhchb                             
#bjijjigajaidcfhjbghccfgjhifccadfgbdigcb
iejciaibeib                             
#iheadiffdaddihebahgcchdcgjabdeagdfgdbah
gabacaajbje                             
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gfejjechbag                             
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afjfadacfgj                             
#ahcicfhgjbhcjdeddcdfghjecfdiieffdiejeca
hjhcgihjbjg                             
#iejcfehcebbgfbbjjfejfghfaggagajiefgbjje
fbigjcjbcee                             
#ceiccgchfiggahcbhceccgeegbfbcdddaeffdia
fjjegfbffff                             

#jdaiijhebgfghigjcejdaicbieeedefhgdecchh
dbabacjdhgi                             
#hfbjjjibhgjfaghjjgfgheibgceagechfebcfgg
cdhecejecjf                             
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dciibbbddbg                             
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cdajdbhjigh                             
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agcffhacefh                             
#ccecafhehhddjbffgbfbjcaiffjjgheaidijebf
ahaijafacee                             
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ibghghhdbbg                             
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eibejaaefcj                             
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jecaejbbcje                             
#ebbccegffcdbjhidcajbchehfgjfjfcjdcjbigg
aihecbbdbab                             
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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                             
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edhcbheddch                             
#dcaehaaiagcdigbbhjagjdagbdeiegbigaffjhj
iebhebbbajd                             
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cefgbjeihei                             
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hjijdbjffbg                             
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cfaehcecjai                             
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iejjccdedbg                             
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adehjahhdbb                             
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ahegjificag                             
#ejghadccaibgfhceacjiaaihgdidaibcgfegjee
cbafjagfbda                             
#bbfacifgeiegjcddhjdijcadjgbgegigggffbhd
caejefeeabg                             

#gcichedfcbhieigbdbaibibfdfiiajgheehcbjh
higabccjeai                             
#bbhiggagbfjbdhaigjecgaifhcbciahhjajbfgg
eehefehdebi                             
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fgfffahjehc                             
#ccjfdihidbciciidchhcgiifebifcbedjajaigi
ifcdadfdbhg                             
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jdjgggceeia                             
#ihibdajgeabcabfjjbjdbdefajhieibbiicacih
bjfjagabaca                             
#abcdchjhifgcgegejadcaijacggchajehhadbjj
hjfhehcbfhc                             
#dhhfjhceiaigcieehfdhdecfcbjhedhbicebfjj
geadfjeecgc                             
#aedjefhjgiaeaajajdhfghedbjdejbdcaghdhab
aggfdjfiffd                             
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egbffbbdabe                             
#jggbdcfjggbfjaagdeegbhcdccjfjcahajehiij
djghjaiahgd                             
#dijjidibedjabgigdhbhfibdahhfahhheceggbc
fahigaiccgc                             
#hbeidgdijefigjgeecdebjaaecaedjbgcccaahb

hcggigfabdg                             
#agjaejegfbegjfdfeebifgiihigahbbcjhgchad
bbfgdceggdh                             
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bjbhcagedde                             
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gfegcdhhdhh                             
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ajdcccgehbd                             
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fdehhhfafah                             
#jgdigcbbedjbijfebfeaegcgdbjhddafifejbhf
hbaibhifhgg                             
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cajegfjgeji                             
#hbjghdabbjcgdfijiddjcdhabcbffiabebaegab
dccjhgjajce                             
#cghijcdijjjijffdcbibfdcgbeggigejbdahbdc
cjbhdcghjib                             
#ajhddhjeacaddifcgdejdejgbjhbaagbhhebcgi
icidijdhfbf                             
#achaiibebbjebcicjfgbgcficjeicebagfcgbaf
fahhijdaidb                             
#ghddadfijhiicgfdijahdhegacgidbidebcggjd
acjbejajdge                             

#fihhfhedjicjdccfecgjcjbicfgfiebjiefchbg
gaggjdecdga                             
#icbijfiifdgdihcgihbdabbgfigfjidccajgidj
hehdjcajedi                             
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hhefehjdajd                             
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dcagedfiiaf                             
#digdaebjbeahjfddhhgghhgiheaffifffijejah
jfifaagghie                             
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gbachfjcced                             
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jacaadfjjig                             
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baeccgeaedb                             
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chdbcjfgdac                             
#ejfigcibchedhhdaacfdbgdccchdhiiahjchbjg
feabceeicge                             
#eijefcjabdagbaiagaihjhhidcbicbaacfejaff
hiecfgjcegi                             
#eghcgahgadabibgieahbcaifggejggjebgcjadc
eeffhijihgg                             
#bhaefbacichddbcbafbafbafaigadbicbfagjgf

jacgjgdffcf                             
#jjdcigffhdbcbajggbeddjcggibdjgdaiabaggj
aihidbhfaed                             
#eccjaehdbeacahceahbjhaacghaiachaedbcedb
aeeifjhchii                             
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hijihahdida                             
#dhbahhdaahajaghadbcgdeegcejihhdhhhbehae
idbjdefcaih                             
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fiidaijegie                             
#hidceeeehcjgejgjjigbhhcghgibdijfeccddja
bdfhcjeejbj                             
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cgggfcaijef                             
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hjjdcabiegh                             
#gebaddfggifddiifjeaahjjjcdbdfgbebdeheie
cibjcbjdfgj                             
#hhjhhighicagedaagidicbbdihacjhiiefdahch
biecghgaejc                             
#ggfadfcfbfbeheaabchjafjficehjgdabfigchh
ddjdjcjdjig                             
#cijhdehcacgiadbdfbijiaefdhdjieefheabfbf
icajcjeheaa                             

#bibjjehdfdjafeidgfjbhdgfeebdjhfajfjeaia
jbbfdehfcje                             
#djedjcbfcdgiajbhagdiabbggfaecffhjifbjah
hfbghccadba                             
#bjfddhjeeiadbhjhcbihgcdaahdadcfjggaffjf
jhgcajhaidi                             
#hdhdhddibhaeadgfhehifcciaedhjideacchjee
eiiigejacdf                             
#iafgfghhccfebcaihecjecehchfccccecbicebd
ibicjejcajj                             
#hcagbjeddccijejejcgfdacicjgidfaaidfgggj
adfdjgfbeij                             
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bicidbihhed                             
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ddgajgffeai                             
#ehgedgahdehbcdghegcbdcdcfhedhhccghchgai
iefagjgfdhc                             
#febchaehaggehhbajffhgfiaaiigchdchhhgiid
fhdabghcaai                             
#gfadbcjahgcccdhadddaijjhfagjbjjehbfjdbj
gbeachfcgfe                             
#edfbdeifbiibgjbifjhghbibhdeaebbgecfjgfh
acifajciefg                             
#jhbhhadfhjccciibfchdihhcheaeagcfiijfagj

fjjfffcbjjf                             
#cjfiihbedghebcdbgeedjhafifibehjcjjihbic
abhhhafhdec                             
#gicbebbgejdfeighgfhhifccjiegccdajfjafdf
eiehecihdcf                             
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jhfehbjbfbg                             
#gjeabjiedbicjdciadcedgifeiahbfjgbhhigjj
aahgeagcdej                             
#bifjfejiaihejjacficcfbhdfhcidfabjcjbdge
jchigedcehi                             
#jhcjghfabiacdfjhhihcfdjfdhejddhegjaefei
fgggfbgidfd                             
#aedahjcebegcdjeacfcdheecbabiiejhddjibfh
ifabcajidjg                             
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jjcjfbaiiih                             
#hghcahjcicaigjdbibabbdiejfgfadbbijjbbaj
gaebibceaji                             
#cdhejhfgegaijdcccijeaahidficbjcdhhaadie
bidjcjcjhia                             
#fjiadiebjihaccjbfgagegjbdaggfgdggjbjdhd
fefdgiggihd                             
#chghhicejhicghjcbjgjajddcgghefigddjbdag
eecgjdfegaj                             

#ihdfgeijiahcjbbaeigefhdhfbagbajachgjhhb
ebedcbhijfi                             
#headdfbibahaegbchjechfagibcfcefcfihcehg
jfddjhjifcb                             
#iaafehefceiefadceichfcdfbhbbchaifabbbja
cefedfiffgc                             
#ibhhbbjggahhhejhcjfaiejaiaefdebjafbihee
dbachfdchfd                             
#ccgfaiicadechcfeahfhiahbehagffdgjbgjfbc
gecfgchbcgj                             
#higjbjfgbajgijjfgicfjahefhbjdfbiaehggbb
bheiefjfjdj                             
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heigbbddddg                             
#ghciibhaefeefbjhhecdhchbfiddbehadjjfiac
hjbdgcfadfb                             
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cgjdgjhjijh                             
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djgfbfhdgci                             
#hiaajfabcibfaddjbjdecahigibghbjdhjgfahd
ihjigdbahef                             
#ajdaideigbbafdbbhahifeaigcadbfijbajecce
chejghbfibb                             
#hjcbfhaijehifddhfihibfaggecfehadgjhfehi

ejcahegjfgb                             
#ehdcibdiabffdfibaeefeieibadfccehdagabea
bgjgheefdji                             
#gccabggibccjhcddaahjjiagbaaicadhbfjccaf
dgbddjbhecj                             
#iecaaccdafhjacfbgfhcdjchjfefhdciefbbdij
ajfdcgjdidi                             
#jhfgaafbcgjajacbcicfcccedjccacjcdhbidff
abddbbiahcb                             
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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                             
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aiibifaeddc                             
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ffcbabijjcg                             
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agcdgchbebg                             
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efgaieigfcc                             
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hbiedghjidi                             
#fcfhjdebidihcgdbefdiebiegeaahjjdhfdcaij
icihebfcbgc                             
#bcaijeadgiiabeadfcadgddhhggcjjbbjdihiff
gjieejadgbj                             
#hciidgaiifjfcffcehgecefgcghfbiifefhfiej

bffgjiedfif                             
#afhdcahdbbdgbhdedeibjbddfjbcibhifajgjbb
hbgajcjceag                             
#bgchgbgfhhbcdhdbffebbebjffddjhhggabibhd
afaehjdhhfg                             
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ifaadajdgeb                             
#fedhajeibbciadjbhagaigbibihjcjehhgagicd
ecbebbjjcbb                             
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fejdaiadaac                             
#eigbjgbahjcjgahajhjdhbhfehdhabddjigjcef
cgfaagfbihg                             
#cfhhbagjheafhjiceahdibjgfdaficaefacachd
jeaaecbdibh                             
#fedjidbfbcjadedafjacccdffbbdfdgiebagbgh
jfebagejfce                             
#abbgjchbihiebagidecdfebiiddecbjiecbfdca
ajhgcaijehd                             
#fgcehghccefbdbhdegijbhahijecbedbcfgjihj
figgcefagij                             
#djbagbbccbbfebiffjheccdefgjhgcgdhidedgf
dgcbeibbhfc                             
#ghjdjdfdedchabccdaajhccjddaabcachhffjej
igdegehfdea                             

#cijjchfcbfbehecfgefcafggcjcafegbfjffafd
igadgajdejf                             
#gjhfcahajbhgejcebjhcgcbdabdidbjibeadehf
caihgfjbiif                             
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bebbbfbdcig                             
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chiadfcabib                             
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gbhaciideib                             
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dcjhbbdjfbi                             
#efjiaibecagafdebbfbchjdfadbhccaihicjjcj
cghhfdbafid                             
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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                             
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iifdcfdgefh                             
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abhegacdhch                             
#gfafddijabbagfaiehdaheffefeaaaeebcjgjif
bagbjigifgj                             
#gbjdfaegjgddaafhiafhcihidgdbdhjeaccfbfa
jbbciddabjc                             

#ggdjdeiedfhaiicdiaieddidggaaihbagchebij
dghhegbfbhg                             
#cbhdjabgedjheibbjfjaebjdbgecdhjhheffijf
dechaccfjfg                             
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dhagfagaehg                             
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fhhghffecif                             
#jbahgigebeggjddibfghfjaaagghihgfgbjbjjj
igghgchcfii                             
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ggdcdihfeif                             
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jehieeadfga                             
#edggibgccaebabdhefdhiaaegaghgbbgcgdeijd
adgabaiecae                             
#bidddffcbbcbjcefjeeehgcggeffbebachdfgfe
ccjejahjgaa                             
#gffiibijecacggdbjgjiebchbfagccdahahfbgc
aeffjdacbbb                             
#ehjcbcighbgighcbifajdhbebiafifaaeggdcbj
cgegiafcgif                             
#djdfcegjigegiagaiffdjjfjidfijiadjjggejj
hdifjghbiah                             
#gecbibajhijjgeddbbigiijhhbbcgiibdbahifb

eijfdcacife                             
#cdjjbhhffhgbhdhhhcheeafieigjfdedefbhgga
bfigbfgecgc                             
#hifaehhfcbcjdjgedcccidafcachdeddjjffhde
dafeahjhbah                             
#hhccffhigjccfadbbijigdbhbdceefjeeeghfgg
cbgdhaecgdf                             
#egagdjghafdjciafcahdcidgfhdieiaigjbddbj
idiifbgefdf                             
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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                             
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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                             
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daajjhcfjii                             
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jhebaaebaeb                             
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dafggaiedjc                             
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hjbfehcghhf                             
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hhgfbjabaja                             
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abcdefdhace                             
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jibgcgdebdf                             
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edifafjjafh                             
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edhgahdaihh                             
#ajjgbdhafcicbfjcddaadaafhfhfidbegegfjee
gedecjhhihd                             
#cbbiaifcjdhbicabciijbfadaahifihdcibidai

aiggicjdjid                             
#afedaihddhahagjehiechahebbheaicffgcjacj
hfhbidjehhi                             
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chifbbciigh                             
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ifgfbachigc                             
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eccaifebeca                             
#bffdidiajiidiaaefcidehbedfhejbcjjeifhjc
idbcjaafddh                             
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bafjjgdjeih                             
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hfbgafgdahi                             
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gbhiiaiibfh                             
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beibfafheec                             
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hfbggeiidif                             
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ihcfcddabef                             
#egjhhhbhfbegfahahbbbaibiiejgdhaghchfica
jhfbbbhgdhf                             

#ggeehiafcgfdghcbcjhhagbjbggbdjgcafhfaad
eebceaegfeh                             
#eadgbijjabihgcadbiejbbgfbaiigaahhfahfch
jhciececaca                             
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geagahhefie                             
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fihhjieifjh                             
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cagcjfihehb                             
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cfjjjcbbhah                             
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chfbbjgabhg                             
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gjjhhbfcbic                             
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jgejedjeiah                             
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efjffieeegc                             
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iegjibaabji                             
#cfaciaghchjaeaebihjcebbdaedjddajdjfdhij
iahhagihffe                             
#ajefhfiafcifdiaciffggbgbfffcdadaffgfdgi

abjgfgdieca                             
#bhfbeghiiaciaihfhfdjhfaeefdcfiicgehjdhh
aadbjbbgbgc                             
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bdajdcjcefc                             
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ihfhgbbjecg                             
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hcihehebfdf                             
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baeijedbbfe                             
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jggheechbeg                             
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ficadhjhbej                             
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bjjigafgcec                             
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ccebidhdbjh                             
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jeeebihfaif                             
#ejbdahidhbcbcfcgbigbgjcgcccegbcidbfajdd
fbgbcdggchg                             
#ehhdhgdfiecghbjiigiifcjghbebdbhhgccbgcg
fjcgafbfcjf                             

#jdbidbedejjddfjcbgffghgbdgdgfjgeihdcdbg
dbdbabffjjh                             
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ceihcjjjjaj                             
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dafdjgebajg                             
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eiadiegifbe                             
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gghhifcgcad                             
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dhffcahbhab                             
#eigjfeaecbiigjjijciecfehaiaigchdeidcaed
geeaafihdai                             
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ichigecajaf                             
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affbhjjejha                             
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gbhebgbegca                             
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deeaeaghhbd                             
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hgiihfcaehd                             
#dgaiajcdehbcgcbbhghiigabfedfiahgggafiab

hhiijedjffe                             
#eiiecgihheaabfebheifdeaggdiifejabdbhaeb
cdhaghejaah                             
#hgbbafhgfjhebiiiifhhgchifdgcjhajgijacfh
dhgedcahbag                             
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hefbgghgdfa                             
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ghjdhaajigf                             
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jidigcjicef                             
#cjcgffijhechgdacbidhajigihigjfibjcaefjc
hhfieecdigf                             
#addeabaciigibcfabbbiidiehgdcbecahieibeb
fheibahghgh                             
#afjcbbgbhejcigfbhacfcchgjdcahajibchghch
chhebhggbfb                             
#cbdbjbccagigjgchhhdiegiciegeaiefibggagi
hieeceheche                             
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cejfghaajgf                             
#aedjbjcgfahechaadbjcefhhcdhhegfdcdfabic
jdefgbaadbb                             
#fjfjeecdjbbahbcjadjgdhgfgjajehgcijffdid
feecdgbfdgg                             

#ijiafdgfbgifigdgccfgcbfaaaacibajhicecff
afhfgdhbbaf                             
#ecgbjjbebihhfejhfhifeffbiaeceiifafgcigd
efheejabcji                             
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bedhhcegcbc                             
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afgddbdbdfd                             
#hhhgcbhfgcjdfgaigijbffhfhgechcjhfaceccf
gjdabfbgidi                             
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bfjifedaeei                             
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hjhfjajcfhc                             
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eaeiiedjdad                             
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bhigfjggafh                             
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bfabghfjefa                             
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hecjbhhifcc                             
#aiageiediccfjedcfcecfihiciheabibdcgggbf
eadifgjghha                             
#bggebibbgiicacjijhdgjjdjdbahgjjhiadgage

dccijjajfee                             
#afjgibiiicbafgdfdjjidhhbjdbhhfgbfbbeefd
cjgbgbadbdf                             
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bfefjebfdga                             
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edfhgbbbaei                             
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gjbchihejdb                             
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aacaebchdei                             
#iffaichbghgdefjbaihhachbeadabcdjjchfaga
ejjjdegdgdc                             
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afieabjjddg                             
#gdbdihgcacggehdgejcahcaiaihjjbiicdegdig
edaiaeghdje                             
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afagccccgcb                             
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cgfjijgbdgc                             
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cjgjhjhbgfc                             
#hfbeedjjddbbjdgijigcjhifgijifghcecfdech
ggghdfgbhih                             

#chddacggjbjfhiehhihbihafgdafcdbifecichh
fhgfjeefjaa                             
#icicfdbbbgaijjegcfhggicajfedjjaaiebgdea
gdbjhhagfaa                             
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fgcigicefab                             
#gdgdbijeijccbdeejcchdcbfjhffhaaaibdbhcb
icaahhhfhec                             
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bbiadcgdjjh                             
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bbgdhejebaj                             
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geiifjejide                             
#degdbhhjfjeeahabfhfcffjfjcichechigbbcjb
efbhdeafjbi                             
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bfhbiecjbhi                             
#baihgahggbcdgdhiaabjeahagefffjjjdjbahdj
acgffejfabi                             
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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                             
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gfbigdhbidc                             
#hiibafjhbchfhggghgchebfhfbhhagefdicfagj
fgjahdjdbge                             
#ahfiafbhciefhdgibbccaaeieiefieaiddfedhg
dciebibdddd                             
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fibdaifjidd                             
#hcafgeehbbjcicegdicegdcgfgcgfdihfhefebe
afehaibdbfd                             
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cahgejbbghc                             
#ebfifejieahegegbcbbhcibbejbacbjgfjjcaje
dceifgfchej                             
#acbgdjjifchghebhibfahibiijfabcfdfjhchaf
biieadhdeaf                             
#edeiaaiheehbdabhhciihgfiaefeijhacffdjjh

hadjbccdcaf                             
#bajacagdihgcbcihjfahgbddbjbbhajgeccbgha
hcghfbjiefb                             
#ceihfjiieachibbeibjcbgehgfgejhbgbegaeij
cgbfbcedfha                             
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jgehdhhhefa                             
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fefbfhijcca                             
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hjhihgeajfd                             
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bgdfhfgifdi                             
#bbhgefffidieehdbacedgcjfcffeaceaigheded
ejjdjhbdjgc                             
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cegdciggiac                             
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jfdibhicgcb                             
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bcdejfcfdei                             
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cjbchbcdgjf                             
#bcjeaaajdgifgcdahigfaiadcbfiadhhggjgich
gccbjcdecbb                             

#iebfeaihbgdagibedffjhfeidffgjcjadjgfcfe
idedjdfeige                             
#hdiffbbciicaacciigahbfbfbejbeidfjiajjgb
cbabchjfhah                             
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heidfhdfefc                             
#bdiajcihaidiabagidjhgahibiidfeciiaeciib
facdbficbgc                             
#eifggfcbjbjabcdajaajcddghjejgfgigigbdgf
fgdaaiahaha                             
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ebjajcbhddf                             
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bjhificfdad                             
#dhegdiceecifgbdbhhdiagbdhciifiehgdcbbjf
gigdiffjgbg                             
#jjbfahiddadbfdgffjahhdaiijaajjdcdfdcgdj
ejfgfgfijbd                             
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eiagiiaefif                             
#cfdighaadfcjcbcgdcbbgiebaaedghgaehgidab
eeeaheejehg                             
#cbcciccadiahidiaihcffgbihabjjbbigaeihee
jdjgfdeagij                             
#adfgibfbgbaddegdehgbieciadcfffcfafiajif

ifgjcbhiefa                             
#fgfagddhbbjfgbgdhgchfgbdbgbgieagghabccc
gjeafjhihhe                             
#ibifidibibddddigdbbhcjciibgfdgfhbhgeigc
egehajefbhj                             
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jgjfbidchaf                             
#diiehgiiibcgcehdbaggdbbbcichjhbbgaiabgi
ehjabjfhbej                             
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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                             
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ebddjieecif                             
#bcgbhaafdaaieghjbddcbfijdffhghdfiddefec
fgdfgehbggi                             
#cibdfgcehihgihebhcjgdgaiiefdidedebfidda
ecjhjcehefd                             
#ijcejjbaebigidbebfdcieieifcihiaidibafje
iigcebehbac                             
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decjcjcjhad                             
#hifhichgbdgcjacccideijgbjghjifggbdgidaa
fdijciijbha                             
#icaaeaefbhdigjbiajjidacfdaahiabhebiacjf
gefiajigade                             
#fiaebajgghijibdadccaggadaebidjfdedcbicf

dcaeidhjbii                             
#cacdfgbijahdcjacijcdddedgjigdaceebaaaid
biihghebgaf                             
#cjadecdbbchebacfhahiffddedacdcecdfbgddd
jgibjhdeagh                             
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ehebcdedfcb                             
#dbeeiageefigfeebdafegfdhcidjfgjhchfaggi
djiggajccff                             
#ajagfbjiegbddheacfffigdcfaihagdibicjdji
eabdeifcccf                             
#ceaafghgehhfdhdddiiehegfijcagdfjgcccabc
hjdjejbfbaf                             
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hbifibdaeah                             
#bbegbaffdefjdgieggcgbgaigjcdhcfbccegjhc
bijhdebacgc                             
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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                             
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haheibbfhhg                             
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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                             
#bchahehibffefhdjgegibjgaaccfigicdejebab
fdhjgfjfebf                             
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cijfgcebahc                             
#eigjjehgjffhdggdbffifhidjjadfdccihehhfj
jdjghbbbbbh                             
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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                             
#jiejjihbjgbjiedebbggijheaegbijdbifjjcdh
jgijjggidai                             
#fihfjfjbhiceihabbgecdfidifhbeaiddibffjj
iabgfdedigf                             

#ebaajcdaficibajeddhbhgcifjcdacajfbdjjag
achjabbdgib                             
#aggjeghbadchafbidiiejggcfjbdhhaebjicccj
adcdhgeaghb                             
#afgebjchdbhcgdbbhbifajcfahdjajeidebcgei
hbjdjcdeaeh                             
#bgehhcajfhhhbacjeahebefeebahjafaffafdih
egbjjgjebai                             
#hefhcgebejijijcaafhejhdcejihbdggjcejbdc
hjajfggeicd                             
#bdecedggiagdidadjjgacjhdacgdgfhdeggjibh
didjiifcgdh                             
#gjajbfbcaagbbdjcgaifcggbhhbfdgaagecajdj
ggjacijcdgb                             
#dbcaaeggedghebhfijadigjfeghdacgjhddcddc
jbfeiiffajd                             
#ehdbjhafebdcebgiifbbiejdjijiihdbbfhfgbc
fgbgbbefige                             
#biefjbcjdidiefedfagehacaieicjbfdbhiiied
ddfjjjcdidc                             
#faebejeehajbicaichihjcdbdbjadbccjifbegh
ibcafeeibgj                             
#acdaibdgabchicjbfeijdicdbagaceiffbigigj
cfdahfcjdjj                             
#fjcehefdiihcddhceefebfjeaajjhiafffibhgh

gcgaafjaaaa                             
#dhgfffgfjadeccfdcegfibdahdfgfigeahahdif
faacdicbjdf                             
#cagcbhjhefegbfgjghificfibbcedigbjaeagia
cahhabifjbf                             
#ffffhahcbjfhdajaefgagdfbbebjeabcaceihga
cbjajajhdii                             
#hiifbaaihhaibaaibegjgifdhcdhiiabbjeijeh
ehfccjdcaaa                             
#dcjdbifcjiicbddiddbgeahgjhfbdjjbicjhfje
hhghjcajjje                             
#aeicifhifjdfbdgceebbhabjieicidggdjgfbff
cgbcbebfjce                             
#cjaiahjaifbgabacbdffjgcehabhbdcgfeebahe
aheciejehca                             
#ffjjcbbgfjciadefdcgcceaiehdfgcdfjaabeed
jdchbdhgffe                             
#aadcffacgjacfdghaedihbeehidgchcdiajdieh
haggjdbdhgf                             
#ahahicdfdhjejeahbfdchcjiafgahbhhgbjhbjd
bfbbggjbccc                             
#efjbcbbcdbgjceeiadacdaicbdbfdcdgeeggjic
acdedidjjhe                             
#aajcadebjhhjagfgeffdehehfjfdhggejgafgbb
bcacegdgcci                             

#dbgjccbgjgdifcedjajjhibbgfibdcfhgfcgiib
ifeibaibdga                             
#gjdeajacdbebhfgdedgiacehdfhggeajgjfehbb
edjgcgifghd                             
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dbabdgdgijf                             
#gdeihhdjhfgadadibcgffdgdbghgdfeaaagcaeg
bfcfdgjihgd                             
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hefieifcheb                             
#igihgfdbbjcccdbfjfhjcbjcdiaicghdjjjihbe
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                             
#bjdjbgccgcdjcifjhadgbfdfgcidhjaegdjfdjf
ajfbhahddcf                             
#beagbbgecgfhcejjigagdhcjhdehjdecgcgahgb
dgjigidjahg                             

#hbcifgfbicifgigcffbfaficgffbhahhhjeejgc
bijjadghdfb                             
#gachdhhhhaejbjbiijhedihiggiiaabhigdjfgc
dhbigaadfba                             
#acaeabgidhajaihbbbcjhcghcdaegdgihdfeiif
adibafgbeba                             
#ddbcigfbjgbhhhbhbbbeiieahceedbgdbgadgbf
dfidbacbcaf                             
#ddiafjjibbgjdfbcfebgiageiicjhhicgcbhdic
iecfhjbiida                             
#ibajecdeifjieedaiaedacjjdhbgiafebjcdabj
chbaejgcbfe                             
#biffdeaeacafgfibbabhgfadeficaafgejgadhg
higechhffeg                             
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cefcfhhdfjf                             
#gaabaeabdjedgdggaagjfjjdgefgchifdbhghah
ggbidhabehd                             
#ihaicgjdcebehbiejgaggehiadcjeaiaagfbggg
hibhgdeafid                             
#bhhcigiaicfgchgiddfedhfbcagaehhfbeffjjd
fjfjcefefff                             
#gdcgfifgjggdehgfahfiiigijhbbhafhhgabhgh
jhdjhjhjjce                             
#bhjgjhiijjdhdbgbfajcagheadbaagihjgfdeji

fbfcbdbjgba                             
#aifhfgiiehagbebhgbfegjiacachahegjdbdhbi
ebjaajieghd                             
#checiebbehfaedhgjfbifcijccdbghhibedgbjf
eejchdchhid                             
#eijjbfifjgfcdjihjgfeagiiehgiihfdadjeicb
ibfcdhedcaa                             
#gfidcjffbhiedbdecdgcehjafccgggdfjdecfad
dffddchjcca                             
#ahiecjfheajcebegihjiefifiiffigghhhbdgdi
aegbgjceegb                             
#fahhfcegjgiefedfeebbdiieiagbcidbfhfheie
cdbdjjcebib                             
#bfbbhegbhfjaefjfjjjjgiibfbcjjcgaaeghggc
hahdjghbehg                             
#cbjbdhabcghdjdedhfihfcjcehjafceehbdadfi
iihcdhahiii                             
#dfiadbfcbceejaadjdgfjdiechidfbgdijceagi
ajdjghiicje                             
#jejbdbgacghaihhgfcbdjhdijdeccfcbebgfjfe
bfajbdidgja                             
#ecgjdgjegbcaigfgedaagichegigegfcdeagacc
cighjfheceh                             
#fidbfdadadjafcghecjgeabbifecacbbdffjjii
jagjhahedfa                             

#fcdjedbbadfhfhaacihhfdcacefcgjbghihcjjc
ibbfiggccec                             
#abjejefhdiagedffabefahigjjjaaagidciefgb
jgccehheijf                             
#ddcdfifaighjaabggeaaccdcciabjcedafddhjg
dcieiedaffa                             
#bhifeddbbcjidcifdejgffcafghiehfiedfhaih
cicicjjcgba                             
#hbfdabgfdaigjaifedhjieijefdeeheihecjcdj
cfijcegbafj                             
#efbcaiaiiicffjiagabcjgjhhfhahhdgibfgggd
jecbbbdbhjd                             
#hjfgjiehgjbacebjjiaejcdceebbjjifcedfdcc
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                             
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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                             
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hfgebcijaad                             
#ahfhejceeedjbgfeadbefgfggbbgjhibadhhhhi
eecgajddhdj                             
#abdcdjfjefjdgfhdddedagbadddgajibicidecc
fedabbciaia                             
#dhhfbecafdfgjidcjfebaecghagdfgjedfijffj
dhhgjdbffgh                             
#deiebafbdcgihcbdjadhceihgcgegbdaeehjefc
ddhahifjbjc                             

#iccbbjdhiijchjdaabjedigbdfcbibiiigeeceg
jccbcfeaigb                             
#jjjeffbdfafbijeddgifjgjdhcggedjdjjhfbbg
feggeaijbdc                             
#faddjebjgggdceiaidhcffggaagbehdjjeaajdj
fjbejbcfceb                             
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jdbifaajefd                             
#jahfbhcjghhcabahffbfifhfebidfiabfgdigfg
aeajfahihjh                             
#bfdgbedbebhijjfjbddghcbchedbgbggejdhbjj
eebcihcgfif                             
#cbghcgdechjdajdjidihagdcddhacjfegdffdad
biihhhhibbb                             
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eaaeeidjbbj                             
#aiejfedffafacighfbjeegdceaghfggegjjhfij
fjacheaghhb                             
#egdeaajcffbgibfehddefadgaaagaifgfadgeff
hfjjfjadgjh                             
#acgefhghibdgceiejaaecgcgabdcggiihgcdjah
eiajhhigiee                             
#dbjjaggebafcehicbghgihhdbgdgabigiihbgaa
dcbfgdcbbgb                             
#jfbabjdbiihaifcaibehhhdfghcgbadcgfghgjj

ddifhfjijde                             
#fcdafdchiccjijebdabehddggiabaeijidjifcf
igehebbjcaa                             
#eiabjhaafdeediebicfieieegagdiicfbddceah
fjhacdbhihg                             
#aafehgbfjfefegceeacjbehgfajghdhjdhecdfh
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                             
#ccdddbhfbbgjdjciheiciebdbdchjfjafgdfgfd
egdjajaehfb                             
#bafhdfjhhfbgidccdaiibjidebcijdfijjeiida
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                             
#ahaihdffehacebhciijadhicfcjcbgajdhhfajc
jbgaecddhce                             
#behcfijjgaadfgedghefccabbejibdejibcdfhd
gdfacjfefcb                             
#cheecdceihhefchfiiiffjaefgahfjehfabdgde
hhihhbegibe                             
#ihcjdidfccaccaacfbdbejdaaejcfgebffjeidc
igfdfeceibf                             
#decbcbgfccbbccaihfjgeajahedgfcdaicabiig
bhbaigbhabi                             
#cfiaggfecgcedjebcbabbbjegbcdfhcbdgejjab
iibjgjceajj                             
#ffegfbchdgbgdjbcfjchdfighheiiegbbebeijh
debbffjfbfi                             

#ahfcbhbbaadehjgehbhaiibejagbcjhejgaaejj
babbjegfdci                             
#dcfbjgjgdbghcdaeehifjbbieegdfdieceaiade
jhefgeabhgd                             
#ibigidjefgjhfefecihdihgagdddabdhhhagccd
ecefjheeicg                             
#jbcgdghiceajbcdidbfhbafecddifgicbhejjeh
bbahaeifhij                             
#ejegceiafhigbadciacbbaaejebgdffcbfdbjgb
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                             
#gjgifggeaaejicajacaeafjfhbfhbbdcdhcgheh
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                             
----------------------------------------
-------------------------------------   
,*amp]n[ne con/ant1 ! natural integ]s   
concat5at$4 ?is is a ,,normal numb] 9   

base #aj1 ,ref3 ,d4,g4 ,*amp]n[ne1 ,!   
,con/ruction ( decimals normal 9 ! scale
 #aj1 ,j\rnal ( ! ,london  ,ma?4 ,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                                    
----------------------------------------
-------------------------------------   
,copel&-,erdos con/ant1 ! primes        
concat5at$4 ,prov$ to be normal 9 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                   
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ibfhiehdifaajjejhieid                   
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cdabcihgeefgbegihgghd                   
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ffgjcfbbaeeebidjcdeih                   
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giehifiibdjjgddajceae                   
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fddjibdijjbbaecebegbj                   
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gghfbcgfbaiebgeaigeei                   
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cehhbgjfiedhhbfbfchbc                   
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jjbihccdhhaiaeagcehfa                   
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deicgfaicifgeedaejehj                   
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ceifbgidadgieghdaehgg                   
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adeiibaehidgfchcajihj                   
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acgfcghfgbgfaaefifgghdaddffcghchcigabahb

jjigbfjhhfacbgbjagjab                   
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gbcefhgejibcfjecibiga                   
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ffdaibibfcdihfichbjaa                   
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afdjigjdhcjcajfgjhibb                   
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ajichjhfeghfhegbicdfj                   
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dfefbecfehjbadhdgabib                   
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eiecaabeabaejbbabcdge                   
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dfhfcbbhjiebfegjeeibi                   
#jdeaghcdjfbddgbeiaiaijaaeihdciejccegcga
bjeifbfbfbjcbajdcbebbacgbahabeebfedcccgc
igihbebjejiaedhdjhibe                   

#jdbcacbfhcjjidhcjfhfdcfbhhghfbjbgjbfbee
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hcjgifiediacjgjiggjjc                   
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bacgihgigijiiiiabadhh                   
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ahcijeejadjhgjhcbchgf                   
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iacdeahddccdddccjbefh                   
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gdbcfgdagfdhcgbjfjieg                   
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icdfgdiigjhachhfifejh                   
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aaciedcaigeifehcdafig                   
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cheabdaacieejjbgighcgehbifbeiagejhbdfhjd
bedcdeddjhhibgjheeaeg                   
#abfadgfhigafddfjfbfjecgcdeagjdhhhdaffci

acidgfhjijbbbgbaaedccjcjaciacggdcdjiiggg
fcdjfbibfgahaadaiafbh                   
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eeagghihbcgedeeafcbha                   
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bigjbjbcjbjdgchhbedhd                   
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ebfgfbhadahfahjaeideg                   
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cieedheeacjjgjibegead                   
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fffibfifajdhicjbjfafg                   
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cijgghabjhahighbhbgga                   
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fiijdcdeahjfjhcibjafd                   
#aegigegbhijgiicbjgbigcahfeihdddaiabbjbi
fbifddbbecddcdbhfifaejhjahfgccaebhchihjf

eabahabacjijbcaeijfff                   
#eccfjfcaigeffadedbbiggighchghjegeiafeja
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gcdeeciaajeijheihgahc                   
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ejhifaahajghfbhdfjagd                   
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fhfffijiehiebbbjcabib                   
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aaafgdcfjjidaacdedddi                   
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ijefcaaabdbijibfbiach                   
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gjdjicjbffbgedjhhefid                   
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afgcdfffbdihaeegchfcf                   
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fdjbeeabbfffhgjiaaaehbajdicgjehefagahaie
ffagfdegihhdcffbhcfhi                   

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gdcafidbgcdcbbahgihae                   
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hgdihejbfjjffihhcghfj                   
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cdgihhdcchidjiccfecai                   
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fhicfeihihidijfbdcfie                   
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dibeahdefcecaahhebhjd                   
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hhfiijhfjdiggciajjdbh                   
#ijidgbjgifgahhdbjeacgbbiihcfeefehaigffj

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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                   
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caccibdeahccddajhjgbi                   
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cdjfbghgcgjbhfchbbheg                   
#caeigadfchjhheccfacdjhjhgddjafgbgehbejc
ffjcceegbffbhhihcfehhijbbcfdhggehcabffhi

icebehhjcdieeefjdbcdg                   
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ijaebcadhhcfifgigeibj                   
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fegfbiigcciiiadgjibeh                   
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hhhdbeicacafebeigcchf                   
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afcibdgfbjdbaceifadjc                   
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aeicedjjiebhfiicjdbhf                   
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ggbdedchibhgiiehhhcje                   
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cdahdeigheefdficciefi                   

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iddeeaicdaheajcaheggd                   
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djhhajihgbchchfgecabg                   
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dechedfbiafidbcdidifi                   
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bchheidaaacbifieffeaa                   
#jebajbgahehaheccfchehegbcgggdhacbbjejcc

jdgjfafbbcgficgbfcdaejcghhbjfdjgdffagcbf
jehgjfdfefccbegifeeib                   
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fgifhdgbeafjacbfcjcje                   
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abfifgdhbaahbhjdgacdi                   
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hhchaifhgidaehfjgfjjb                   
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ebibhejajjgjcihehbiei                   
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baibbfibgjeegdfhbchce                   
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cijaejaiejfbdbjcggajggadaggfdhggffadahai

jfbechhceeaaihdgbcdhd                   
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hibhabffafggeijadcjef                   
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edecbfbachhaabaiajhgc                   
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jbddjdbeaijgiaaaedaia                   
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gigeaefbiedffbjigifdc                   
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#gdecbabicjigacbbbgaigafchgbidaeafiidica
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agacjejjibhdaeijbgicg                   

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eabfgggafddejieegjeaj                   
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hiabafigcbjidfhceifjg                   
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caadigdecfgigichibedb                   
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bggidfdggfacgbhficdad                   
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feihajcfcajdbacccdcgj                   
#faifcijjbgehgabeefabdafdhhageeechcbfbgh

caegdejcafbiibfcfheddfgbdcjjcadfjbiajifj
ibjieeefdcgeeccgfebbf                   
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ahgbdaicjajcceaifiicb                   
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cejeeedbjbcgfchgfcefb                   
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cacbadiegfaabfjghibbd                   
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efadjheejaiiehijccidf                   
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iifegefgjjfbegabgifch                   
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ccfdacfbjagaheiacdgic                   
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jggjdibahcjacbaffhjbd                   
#fhaffdcbiigfccfejcccijadbbacgahdghbjccd
aagdfbdejdjhhbjhjbhajefhgceiaccifgicjgdj

ffjdhchefjehcfbfdcgbg                   
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ijdfhdbfhacgafchbdfce                   
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hdifahidcjahagdjghafi                   
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jaggjjdedjjhbigfdhbhb                   
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bjbgechahdgdgbcgdgbgf                   
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fieadffhggdabefbfjjci                   
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ejiebgbhjiedbaejbfajd                   
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fbcidgiijggdjbdieadgf                   
#hhafgceijjijbiiaabechbbffeejacbhhidfdgc
eiadbhbhhdcjaeheehjjeghjgceedggffgiccedj
cbcjhdfchiiiagjdhfgcg                   

#chdiajajdceiddhgfjgibgdbijbfgfhbadjehic
gbghdjjhggdbjdadjabajfdbgbghehcejjebeigi
ddhffcjeejjchbcjabchf                   
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gibabhfgbcjdacajcjcba                   
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hjcgdhadejbjejfabbgie                   
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ebhejfgcbahdjaghgahdd                   
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daiiecjgeejieiiiffffh                   
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iabbffbdfcgbcdfdiaiag                   
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jadbgidfadcjeejhjgdah                   
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ggcfcejbfhjafgiegfeegcbbdfggfhcjdhgcibaa
cfdhdbegjdhdciidfbdfi                   
#gcifijheaddijgfaeeehebcdhghbdaiajedbbfd

eccggeagfgfcfbeaihhhhdcjghcgiegeicgeejgj
hggdceaehabeffddiafgg                   
#jgchdbjacdgdfeaabjhhjfdghbcechefehcijhf
idfidbgeffgigiadhceefbfdhbdgdijghcggcece
chhdaaefaebdcbifcfgje                   
#dedbjfigigdaicgadhfjjhfehbhhbdjfegfeeee
cjgaccfdabcdgjgbdjibcjhbjcgafafjidibdgic
ccdacedacciedejgcbfgg                   
#ahdajbjghbfbidigbchaibbjafbfhfjcfdejcii
ihefgbiicibfegaedebcejjaidbhfbicjafieegd
efjheadhgggcdfgeeafai                   
#hebdjihdjdecfbabefchcfbgacfgedjhjfhiadg
ecfajdhifjgibecieigaafecjaiiifhiiifefdeb
jieagfadhgbjaaighdfie                   
#gifdacfbicdegbfhghdigadacbdiiffahhdiecj
gbhcjhgadceihefbaggjihgajfbbcchajdjbibcd
jgagijdadbgbfegchbfgc                   
#efgeaeebaahaagifjcccjgjjahbjgdcjjgebide
iciceejfadeicejcahgeaghdjijgdbcieggddhid
bicceegaajfdjjjcbfagf                   
#iieieecccjfeijihbgaehidaecbejbigghfdfde
dcdfcabchbbgdgeeigidcagcjcadhhcbabfified
gjaiidjdgdhfdgaccaicj                   
----------------------------------------
-------------------------------------   

,! cube root ( #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 ! d]ivative ( ,zeta      
function at #b4                         
-#j4icgedhbedcaehdcgecgjbegdjidefghfdigg
highfjbhhfadhbiibehhedccdhjc            
#fjddchaacabgjgebbgicfhidaeadaaeaeagdica
achbaafbdafchecejeidjdaga               
#eifagccbdgaigaheagdiabdjbfhhbadddcgjjaf
cicajaejdeajgafjcgcegdhgc               
#acebiefjegacceebeiccahjejeadhgbecdgiiih
dgagcigegjcageejcjbfaijgc               
----------------------------------------
-------------------------------------   

,?is numb] is 7exp7#b7-g7_/b  ,ref3     
,francois le ,lionnais1 ,les nombres    
rem>quables1 ,p>is1 ,h]mann #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  ,!           
,dubois-,raymond con/ant to #ajbd digits
  "h is an expression = x4              
#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 call$ ,gomp]tz 7#ahbe7 con/ant     
int7exp7-x7_/7#ax71x#j44;inf9ity72      
#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                            
----------------------------------------
-------------------------------------   

,! 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 
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eeebbgcfiiagigbjiigegefcghbdeefjjhjgcbe 
#bdgigfgbadehdiheagajcgbadjddjffgdhgcaeg
ggbabjciidfigdfgfjigafhbegajaggdbgjbjia 
#igffbecaiehbfjcicjibbbidajbgbfgbdeddgaa
cdddgecejfafbbhhggfiaidgfafhffiejgajhhg 
#aafidbiidfgccijeafcdafdcefgcgifaafjghaj
ghbjhjibecjicdcafjfggcjhdcefdiejebhbbgg 
#adchiigiheiageeigbiedjdfcdbjiefgbbdecfi
eahiegcjhegieihfdcacdgicchehabdaeceefge 
#biahiiaegebbeccgbjaifibfiedhhjbgfdfjegd
jgfigafecdbihhddcagfcdfcgacbjchhjijbcch 
#ahiefdijbcbbdchdjdediibddgdcghehhieiijf
bejdiiijabeghbejhjfjjadghdbgcbaijhabhag 
#gaaicdajhjbdhfhhidebaddghhjfdjjhebiaade
ehjbahegbeijbciajhibibcdiejhcdefjcheeih 
#bfddacjejjjgjhedgicfbjdfdiihfbefcejaegh
jfahjddjedaghgdbdhacebjjjfigchdhcafeijj 
#haiedbajeaaefdadffbgagjbfefjebhdhjcidfj

ijfiifjgdahcchgddfijfbbfbdeeccabcjgjjjc 
#gdfgfaiiiibdhbaifeceibjjbibihefbccgfdfh
chgcbiiidjdghiciccejicbahhhegijgieehajh 
#ecbajfgdbbhjfffgfahecbfcfadbefdiehceaja
gjbjbeachahahbabfhbfdgghdbjjighdagjbbgb 
#jgdicddiigeghebbccdbjbibiejccjgbiegicfb
gjfbffhfdjcjadhgdbjafgcehajefaaaeaaiadj 
#iccaejbfdaabdececjchfcfahabhfddbjaehafj
jefibijhcefjfdigbfegiaihbbigfbhadghjfba 
#fbfibdahiejfjafiiiifffdfebcfegbeebfghbj
cfjaaeddabjaijjiabhgfdgdcfeeaecdgbedjfh 
#iggabfhbbejdgcdceaajdbchdejdbjhddjbahhe
jahfjidhbbacdgdieeicggaedaaihfgighfidjf 
#ijacfhfafiiaifabijbfjjhaebgighaaacdeeic
cijaafccecijccbjfcjcfbciibfjejicjccgeaj 
#dhgchghbeabicbcahbhfgeegbiaejafcehjicej
jedhhcbbbfgefjhbabgbjcgbgjbahcgbajfcfah 
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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    
us$#bdhchh1 alloc#bfbjif1 "t#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
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#ijghhfahfjfdejgfdhbjhfhbfjjgcgfabcccbjj
ccihbbbfhhii                            
----------------------------------------
-------------------------------------   
exp7,pi7-,pi to #bjjj digits4           
#ai4iiijiiigiahidgegfgbffddbihdffijdddif
jfhicfhdcbbeajfagbdgjajahagbafebeidddjd 
#bdcghdhhhicgagagbedcbaeafichjdfahbhghje
dffdigccdaiihjeadcbecfabiibjhfdgadhacfh 
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dgajbhdcjgeejegejaeggacdfacdeifhfhjafaj 
#gjdfdghjaejgbaagfbdhfcadhdghfjegghfgijj
hcccaajhcbefiecgdfegbiacfhjjbjcbccjdibi 
#fahejdfcbhcaaejeddebbciiijgcjcahjajhchj

fbagbfbfgfiiehjcedcdbjiffehedfhgfddihgi 
#fdcaeiihhjcdceicfefiggiejciigcdbhccacej
jhiegeffhaehgigceeghaccdibggiaibdijhdfb 
#bbcidhifcegdfedfhiejadhiaahiaijicdgaheh
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hfiihjhajejajjbcfegfgigbbdjdaabgacifcia 
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cdcbefhdjibidfhijfafcdfgfgegheaihabghej 
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afbeddjdihgfgibccdifcjhcjbcifiebcgaaihe 
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#aaffacdddjjbebhihbighddciigddjjiefachid
jjegicgdeacfgheidahbfjdbdghhdabifggadhf 

#bcfajfigeaejchhghihbdecgbiabccbjfaacffj
fhdcbbdgfchdjgccedfcibbjdefgficfgddefbb 
#cffccfbdeciicahhabjjbgdeebhhcjecjhiibbh
jbdffddcefaceahfdheieeccdhfbfdijdhacchh 
#geigidcjjfaabeaijifbehfifffghhbgggbajgi
fccbeebgcejbdfcafbddjcefdcjbegehfijhcdg 
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ddghhhgfbfcbchifcihcceehhfcfifcdeehbjgj 
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cadacejabcjdfacajiaciijjgfbhichgifjfhdj 
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fdcjdbfdegfi                            
----------------------------------------
-------------------------------------   
exp7,pi7_/,pi9999,e to #aajj places4    
#a4jcjcdeebdbafbajhcbddaeebdcgeddadbciac
caafgdecedbfcejdggebjfjcgfidcfhehccccfg 
#jghdffecffcdbiifecahfedacgbaacdaabaehfa
dhecjibfgebhcjfgjhaghadadcaadhbagcggdcd 

#dfddiadgcececjegiabagjfdeheagaiebcghcab
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#iahcjgade                              

----------------------------------------
-------------------------------------   
,feig5baum r$uction p>amet]             
#b4ejbijghgejiehibhbbbhcijbhgcbahbaeghfc
habgacgfgbgadiiggccfaibjef  ,feig5baum  
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ehaheeggdgegfhfcbgdefeacdc              
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no4 #d1 #hi--ib4eh;,f#ad                
----------------------------------------
-------------------------------------   
,frans5-,rob9son con/ant4               
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#bcc-bce                                
----------------------------------------

-------------------------------------   
#agjjjj digits ( gamma1 z calculat$ f a 
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gjebcigfabcabchdabefdcdidaaghidihfaehbh 

#eigjbjigajigjciaifcebafhabbehbbdgefbgai
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#djheegcecbbaigjhbaebjbjjjabhghebiadhfbd
ahjgbdcgbgbfeedjbfjfhiaiiggffijieibicab 
#hhdiffeddhbcceacjgfjbehbjdgbbjagihaeahb
ihaecgaaegebadeachidbedjdjfggfdadaejcgh 
#bfaedbhhgcebgdcegbajibfiiahbihgccghdicd
ibejfgiddhhdebhjadbhfaijffchfdhicbjchcb 
#bfbdgeehebiadejhgghaahabfjffjibhgdbeidh
febiggcfgibgdgdbgdcdeddddidbiiiagibjjaj 
#eibhbeghhehhaacijbdacbccfficbcihhcfdfaf
eeicffdcdhhicagbagjdffcbigdfihdgcbjiedh 
#gafechgffeigbeighgdagcgcicjhgciehdgbedg
bgcgcgfbjcegicaafbhaaahibabjbdgijjgabde 
#fjfdficegdagbhdgfahbgjhcceicdhahhibadhb
feibecfeibbbdeadccciicfcieajiidehiidfif 
#idfcefbefcbbidcabcgjdgaegfiagfehdccdche
jbadifggbigfjchdiijeiaidbdjjjiiihidchch 
#gigjhbfjhaigcijidbiggadidbgaeijeiejiddi
hccijdfddifciehbjbbgcdbcjgaheagjihdbeii 
#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 us$#cjjajjd1 alloc#beeedcf1     
"t#djbf4fe  quit bytes us$#cjjaafd1     
alloc#beeedcf1 "t#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               
----------------------------------------
-------------------------------------   
,! ,eul] con/ant squ>$ 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 cub$4 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 transc5d5tal numb] 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 ! ,gibbs ,con/ant 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                            
----------------------------------------

-------------------------------------   
,! ,gauss-,kuzm9-,wirs+ con/ant4        
#j4cjcffcjjbhihgcbfeheigddhabaija       
----------------------------------------
-------------------------------------   
,! gold5 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
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gcbbfcgecdgefffcgjjbbabjcbgebdchiiiggcf 
#jjfjgdjdbgjbigbbjcfcdgghjdhbihhcdheeahi
ebejgidgdfjeeaiidjcdjaajggaafigbefddbfa 
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fcjbaagdghghcdaigefddejahchjdajbibjcbdj 
#djhhbffagcddccijijbfcebbcejejfhbhehbhed
dcbhcifahdhjibecgfacjhbjaaefbfhfiijgiii 
#aagjhdgeeehfihbaejcajjgcefcbdjdbaihhefi
ehdbjjfhbdciibfiecghddjcbjbbbbcgdfbhadg 

#feibcjfjeedgdgficfhcjegfedifggfijdgaaei
fbeejbdgdejghjifbdhcgddiijhjbefacgejiae 
#fbbceijhajajecddicidaggdbidbggjiaddeaff
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hbijfcaefghhjfeahbgfagfegihecejbaffeiih 
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ifbejfgjbgigebdcajaaicggcceedhecgjaafid 
#fgdhehhhhcfcjhjcccbhggciegafefbgecdjcfg
bgbahjgjefbbefbcbfcgdadhhccdiibhiigjbeh 
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hbafcebdbidhjbbeffccjdeheecfcfccgbeagfb 
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cggfcagccegdbjegeciigegdcgcejhdjifeggib 
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efcgehjabffgbhjiebebgcgihbhjcjjgffcfigf 
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fibgdhbbjcgdhhaabbefciidjgebbgfbfdfdiid 
#fehdfcfgdjaieeiigcgjbhchcicaaihhdhbbcce
eciifdighcccafejjhdfgdiabedebbiefeabdji 
#cijifcghdjiedafijabcdfgecgebhjacijhjhcj

hfcjbbfeccebchgjfibgcjgaihdfedfdidedigi 
#ajaacdbhgaedfcffieedcdcgdfbaedciahhfebf
jhecffigdcffecjehhefbafddaafdhjfhiabhcg 
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abceffbagbdafibfhjfbjfdhhdecagfaeafdbgb 
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abbdeibgfgaahiebggccchbccefjdccgdbjacba 
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cdicedeahbdjaabfgajdfjgcjigdabfhabgijgb 
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jihgibfdgfibegdfihhabccdfdbiiebfgcjhbcg 
#djegbjdjfadcgdhgjjhfgjdhfabeiieijaghdbd
igfhdehddgcfhbdhbgidghbdgecagfcchagdhad 

#giiegajcabjccifcdebbfgdcdaeabcgbbcbbded
fbfedfcbhcecefdbdffbgghfdfjhcihgbagiabg 
#hdcjhifdafcfdbbbcgaebhbbaiihfjhejfjjaeh
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gcbihbihbbadegdbccbgcbfeihgihbgefiejhih 
#hdecjfbdjbbcjcfdhfcdggbbifgjefebdabgjce

hhghcjbhaidjjgdihjegedcijafbheghfgdeeca 
#cbgaigfebfjgajgfdcaecaabciaebfjggbaicfb
addcdfjifjhigehgbficdbbcfgdccafacgahegd 
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#dehhgfdhcjgjcgafggcifaedbfeeiaciijhcjcg
eggffcjbacjiijhgabgaihhgjfijcbicjdgjabd 

#ajehfaejfciihebiihadageghjdcjcdhjhjcehh
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#cafgbgjcgbifcfcfgeidbafiiicgiebb       
----------------------------------------
-------------------------------------   
,! ,golomb con/ant4                     
#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  ,gro!ndieck's
 majorant4  ,pi_/7#b99log7#asqrt7#b7772 
----------------------------------------
-------------------------------------   
#a_/,w7#a71 ! inv]se ( ! omega numb] 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               
----------------------------------------
-------------------------------------   
,kh9*9 con/ant 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               
----------------------------------------
-------------------------------------   
,l&au-,ramanujan con/ant calculat$ by   
,philippe ,flajolet ,,inria ,p>is &     
,paul ,zimm]mann                        
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                              
----------------------------------------
-------------------------------------   
,! ,lehm] con/ant 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 con/ant or ,gauss con/ant4  
also kn[n "u ?is =mula4                 
#a_/b99,pi7#c_/b7_/;,,gamma7#c_/d7b99b7a

_/b72  also kn[n "u ! same numb] divid$ 
by sqrt7#b7-- #a4hed444 see ,d4,h4      
,lehm] ,! ,lemniscate ,con/ant 3 ,,mtac 
7or n[ ,,moc7 vol #c4 pp #eej-eea       
7#aidh-di74 or ,abram[itz & ,/egun1     
,h&book ( ,ma!matical ,functions1 p #feh
 =mula #ah4ad4g ,dov] ,publication1 ,new
 ,york #aifd4                           
#b4fbbjegeedbibaaihajdfdhciehihiaaaidacf
hbgedieadcafbcafbhafhbagjchjjgijehgjgjd 
#adbejbcjbieecbifadbijicddfacegebfgaghcb
ahjeefjhiefijacicicefidgjaaaidcdggebceh 
#djdbbfdadigafdijfieaicfhiiigiicbadfjgbc
hcabacijhajbjfbbahigdbifjjhefeedeciggbc 
#jecfiedigajbhhhhhcbeebe                
----------------------------------------
-------------------------------------   
,! ,l5gyel con/ant4                     
#a4jihfhehjeebeahgja                    
----------------------------------------
-------------------------------------   
,! ,levy con/ant4                       
#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 ! natural log>i?m ( #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                            
----------------------------------------
-------------------------------------   
,! log#aj ( #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 log>i?m ( #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 squ>$ 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 log>i?m ( #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 
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cheghejifjfbfjicchiiadgjfabjacjbdcajhbf 
#jeahbfbiehfdjjgfjjcjeididcbaaffhhjddfaj
facdfhdeccih                            
----------------------------------------
-------------------------------------   
log7#d7_/log7#c7 to #ajbd places4       
#a4bfaheiejgadbiadhgdaiijedbbhfheebagjhe
iiagabhjbfcgfjheegdacjihhgfggcgjdjbgfah 
#bifajabbcdecggjihijcdiaabbgjhjcahgffbgd
cjchihdfhihbichgbiejhbgcgbcibghffiiijjj 
#gaiccbhaafidhffbccedijddcabbeaibciigjcg
cgcdeehegjhfhafiijfbafbdagfhchgebahehjc 
#fhcgaiahgfddcgdjfafdjegigaeheacgehahjbf
bfbfeeaedcgefdigiejigiebbhgfjjicihgbdgc 
#bcjgaedjcjcaeiafeafddjfbchdcbagjechdigb
aefiggaahhddjaijhcbgbhhdhghidahhahddcjg 

#iaicdeiijbehidjfcbcbhjceihcbfccjaaacbae
fechicihbaefghfgbdccidjiffdhefjccichfha 
#ffaghgbdbehaihiicgddhfdacgfbaaagjffebij
cfddgicicejbdcdjbbiafjbgchbgbjdgbgebcgd 
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biiddbhhfcghfeadgjfagfgaggaejiijcegciib 
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degcicacjeebeiehcfaddejhbcfghjeicejdggb 
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#bidfcdbjgfieibffjhejjfbcehadfeagfcddiac
fdahddjfcahebgbjejijicdhhhibhggbdegifag 
#aigdfdifjig                            
----------------------------------------
-------------------------------------   
-log7gamma7 to #ajbd digits4            
#j4ediecicabihafddhbbccgffagfhhjbijgghhc
cjfihihabfcjfdgiajijaeacjde             
#gffcadbjjeegecjdgefbfahihiaabgfadjfhdad
ffibgegiaijdjdieebfcaheij               
#edejdaggcdedihdhjghbjgabiciebefbhggbeji

dgjchgfbhafhhbgjigefajjbf               
#aaefdjhhgehhfhhidaiaciigjdbfecghcicbcea
jhcbcgdiecfhgadagibihggid               
#cjjjfihhjcaebijfjjfhcfcibceeahfhaichdgg
ceehchjeeidafhcijhfehfbej               
#bhdaeecjfjgecijgcddcdbfjgijifejfbeiacjc
jfdeicdbcjjfedabcfajfjjda               
#aahfjgdhibagdffgejfdfaebfajjhdcdbaabeih
hbdgedfajhhbchhbdbhecidca               
#fdedbhaehiejfbijicgbheefdjfcagabhfjegcb
hbfjafadbiaejbfcddihgdgij               
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facgffhbcbhgiigiajfgfccif               
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edjgieccfbcchgbjbdegaafdf               
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cbihehidcehdjddfdeacefiij               
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fbhhhefdhbcbhichhgjadchjf               
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cieijhjdchadgbgibdbjbjddi               
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jedabhgjfcjafajehhfefjaji               
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jjgfghijjhciciibghagidcjc               

#dgfcijceadbjchegcjhgcfdcecghdcdgdgagjce
ghdajfiegceigfjbjgbcdedag               
----------------------------------------
-------------------------------------   
,! log ( ! log ( #b to #bjjj digits1    
absolute value4                         
4#cffeabibjehaffdcbgjabdciaehbcbffidfide
dbfcddghcgajebfcjecfgggacfgjefafaecaice 
#bgchedideehbbheffihijhcehcjbebcjdecfdhc
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#dfhaaebbgagjfcaidcbedhijddfigefdicfbbcd
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gdcggiafbghbahaicddaiifcaijbajejdbiccja 
#bdbdaejeibgibfhjigedjefadjbibjiiagfaehb

bdhjbcdbjijahhdabgcfhabbficbabhdjcdiccj 
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ghggdebgjgabeagecfadiahdgaaeeecgdehjicc 
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ajhjhceeajceagedbfegcfihcfabfdcbbjbfcdc 
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#fiigaedfdjfgbdbebjidiabjjbdbffediieddcf
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cidbfihaeefjaggebecdiagaadbacbheeidijcd 

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bhcheifiidjhabcgaahfjjgdbhdbbfhdgjdacde 
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#ffhdffiihhcjfhccgcdghdibdcjicajebcgebbd
jjghibgbaffe                            
----------------------------------------
-------------------------------------   
#a_/b                                   
ln7#a_/b  #a_/b #e   7 & "h is #bjjj    
digits ( x                              
4#dhabaahbejeifjcddgdiggehiacdbdcfhdbcac
eahdccdcheffjeaiffajahafhhdjafchfgfjhbb 
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#hejhhicbagfcffjjbeffffaiecbajfgifhagegf
fahdgcjgceaeihfjcihdhdegededabjefcbcdac 

#egjjdghjjfcidhgbbdcaebfaghifhjjdejicfci
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aaeccbicaddeacdgdggdgeaiegjjdgiigdfdfgf 
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bcjigcfjhehjhdebgeicegicdjfbbgbcfafjgea 
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ihdjhbcggajb                            
----------------------------------------
-------------------------------------   
log7,pi7 natural log>i?m ( ,pi to #bjjj 
places4                                 
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dgbidhabiaecaaegaeacfbcjgadgbacggfihhdh 
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#bebgfciciadbffcdeijbijbdhdggccehhfiicgg
hibjcaaifcjhbdgefgidjaaiafjbhbagbbgcgih 

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jedhgbfchhjbbcbgjjifdccejdieieaahaejffb 
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acbcdbijeeffifddibeheehbfbcaahhbdbbcbdd 
#ffadgfgciacfaecccidadbfdecdfjjhhaigiaee

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#hhdaibegdhhhahicbjgifegaiiadddicidjbecd
hhcbbhbfbhac                            
----------------------------------------
-------------------------------------   
,! ,madelung con/ant  79 absolute       
value74  7= ! ,na,cl7  ,ref]5ces 3      

,ri*>d ,e4 ,cr&all1 ,topics 9 ,advanc$  
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 books1 #aiif4 pages #gc-gi4  ,&re      
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summation =mula1 ,j ( ,phys1 ,a vol4 #h 
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,bailey1 p]sonal communication1         
#a4gdgefdeidfccahbaijfcfbabjceeddcigdjcd
heafadcffbdgdagehaebh                   
#becejgfejdjfbcecbgfaagihijgehcfbfidfjgh
hiicjhcbehaechgecgajeicbhbjbiiddahchbhj 
#acjcficcjjbaefeiicfcbhbcgffjgagbbigefhf
eibchjcgafgbjchajdajfjcdbadeefjfdchbggg 
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bhejhfjefgiaffhafgegciibddgfhdabigjcfgh 
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hibaaghjcacdfgcabbifdihehbhhbhahdcegacc 
#aeiadcagjjediefcbeccdhhgecfcjbfgjdbefbg
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#dibbdfacfbciibjdjjaegejfcjcigbadffdhaaa
eabcgcfdjajbiejffjaaicijdfgaidcgccabecj 
#ddeajbiaaeadfcigeiccaiahjdgiggidfjiiccc
gdfdbidbfefbijhificddggibifhhedaijddjgi 

#adbeehcbgbaiigahdjijfgdfhjbcgfaechicedd
efeejcfjbgcjbheddjhdicddcjbhjfbfgjddahb 
#dabjjdcigdahfgffaggbddgefciecdddbcjfhec
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fiibcjdjhbbadcgdfdeffhjdcajefibffcaigee 
#jdeibbddcbdhjgdhidjhjfbdgdicfajgjicfcji
adibbdcfhihficcadjijcgifhbcbdjgijjdfbhd 
#dhgcid                                 
----------------------------------------
-------------------------------------   
,! gamma function has a m9umum at ?is   
po9t4                                   
#a4dfafcbaddifhcfbcdabfbfeiedbcbegbacbhd
fhaifbjdjjfddfceabieihhdji   is !       
solution ( ! equation 3                 
,psi7x799,,gamma7x7#j  ! po9t y ( t     
function is                             
#j4hhefjcaiddajhhhgjjbghhaeijjehbehhgccb
jgieaeccffiijcddhhgabjjafei             
----------------------------------------
-------------------------------------   
,m9imal y ( ,,gamma7x71  ,! gamma       
function has a m9umum at ?is po9t4      
#a4dfafcbaddifhcfbcdabfbfeiedbcbegbacbhd

fhaifbjdjjfddfceabieihhdji   is !       
solution ( ! exquation 3                
,psi7x799,,gamma7x7#j  ! po9t y ( t     
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               
----------------------------------------
-------------------------------------   
,! omega con/ant 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_/7"o-n9? con/ant7                    
#j4ajgfeciaibbfdhdegffaecbcddejijidgaije
hgig                                    
----------------------------------------
-------------------------------------   
,! ,p>k+ or ,r5yi con/ant4              
#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 ( #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 
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gjcfdcdegehhcdibhfjabcabbejhgbbjbgcageg 
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hcdbcieddaeaffgejbbahhdabcjibbhcdicaeca 
#ejcggedcccbchgbgfdjjdgcifgfgaijgiafebdh

hbdaaeabaijjceighaigfjecdbcfidagaebeijf 
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fgadighbahbabhdecaafajhjifjgfjjbcffdjaa 
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gbffjffcdjaedhgeghbjgjfchddfahghcjegebb 
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ghficeigcddghheiigdaefcihadbjafhdffhaha 
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ifcibaeahggecgahadfcbagibddaaahigcbcceb 
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decdbhadicccaceefdaecdbfjejaicffecibaab 
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jjajfgcbjgbbejceebcijbedcbeeafcggdefgjb 
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jiafdggjadcgdieaegfdaaefdfiajdaafhieeai 
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jgbihejcihibiiefjgbfbahbeghaeadifjaaedf 
#ijiidieghghiedeaefeaajiefcidiiaidefhjdh
dgggfifjadfaafihjiaejibdaiibdaijfebbjbe 

#ccigjahegjdacbjjfjgjibbhfgfegjfichahbdi
dfdcbgaiajdadefabjhgecidahdagbggbdadcib 
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jjcabbjfahagebjhhbjjicibadbcajhidfbaeab 
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aeiibbhhdhacjiedcceehjjhdjiabedgbdgfacb 
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fbhbgjdhhacihiccaedjjhficbcefdfbgaccdjf 
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jgigiaabdchigijjejaedchcejhcecejcdjijab 
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bcgicjgbbbbdcijabacjfbcedfbadiccaecebej 
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hfbfafhchhcbjbeiachjdbibdgiebedbdahejch 
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fecdifdcdjjdfbibifjdaefegjcdajibaaejahj 
#chhgacjijaeadhadhddefbeghfjbihdccgfheaa

edebhibdeeabchadbaijhfdgfcfbbfceghaiffa 
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ddccjfaddgijhcajdgdcgcbbagdafheccdjefga 
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iehhdabahgiaechacihfhegaddaafccfbffjjdb 
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eeaeehehghbegjaecgcbfcbgiecfidjjagfjadj 
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badfdijfjbgdebjcigfgejdjaceefcaideeebgh 
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dcdabcjjhehbgafiahicgahhehbaejgfafihgce 
#hedhjaehdfjegegehhabbfhfeiceiabgjfjcigi
ghbfhgbeffgejcdadahihhfbgdiheiffacjijbf 
#ghajebeiijjgcefbhgfcffeahccaejciegeghad
hfcacjfjfibifgfjdbbeacajaeibigcjafhedaj 
#hicfbaiaegidchejdcdfjjiijcagbehidfdbcgi
ieaijjjgaaicfjjifjgjjahijdgebggfgacjgbc 
#heehgeehgeidihjdceecgadjicaeehcadhiaadd
afhgebigifeiibdfjebfhffdbhfdgfccahbibeh 
#hdhiibaidajjicbcgbhabgjabfiajcacbaibcbb
eccegigggaahahdaedfijdbbbbfjbedheffbbdb 
#fdbeidijaehgjbjgbicejibhdieahacgcfiegji
bcgfdgjddcbeidfgidbfibcaebeaiafhdeideea 

#efhiijccggigedjhcjcehcfajafdjiidjhjedhb
afaeieeidbihfgchbbgjdiccbjdadceceiegija 
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ibfiaccehccfihicbigahiafgdcjggffggigfhj 
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gdaicaeiecdiegficbeihafbagfbgjgdgfdjidg 
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eifcfbihcgahbfihhjggffghjaaecjiahjihadf 
#bagedcjiaedccebjagdbgiebcjfhcdeagdddhac
dbcbagfbceeehhehbebgjdbfhihfcdaeabjcbeb 
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jjehfajifhffdicjjjicdccbfbdaiacibhcbfdc 
#ibbbeeecjdiibjecedaiacacbcehdecjjhahffi
dghjbihjfabfcibbjbecjbgbdcajehbbdbddhfj 
#gjiafeigbafacachbjbghafiehjgecajaeajjhe
gchfdbagccacadbjjggdfhcfbbbacbcajfggjcj 
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aafhciafieefdfibbjfafccfbafhfgigbheggdi 
#bjhfagehbgijhadibjfegijbcjeebgichaaaifc
ffffgaheghhfbeeiedidfbghbhfjgeaedgecjda 
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fiicjdcabbdbihjjfddgcdeihifehjdfggbdaei 
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bgcebfbbaihcjihbibfdaagehahacgficggcgfd 
#djjfgddaaggfgciaifidfcdechfcihccejcajah

gabbgcideeejfhaifhehhibdeidibfdjbjgcfid 
#aafaagicaacebbhefidaeaceedbcfhejhaaaedb
bihfhdedgffiagahjfficbibabcjfijhfjhiedd 
#iheighbfcgejgagdheffhdehdjggahecghdahaj
ejcafhfihhabcibcgbecbcceicehcbaibjfeeej 
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ciheebfgagajheehfadacddcddggbbdfhhajebd 
#gcdagjccfdidjgijdidbdjajfibedciifchjhah
fghdiaejbcfaghbiedebdgfgdghadceichiecha 
#hjhgffbcjgafahcgffhdedbhefebdcbffhjfbig
bfijfbfcghiggagfiecffgcbjebebgageichchc 
#iabfejhbcgidcgeiggcfejgcibhefdehefjcjej
ajijgifjaaehgchfadhfjgjachbdcadaebfgfib 
#ggdbahfeagahfdgjfcjhjhiegabhdieiihjecbe
dfhehjbfahjhfhjhabbcehfegfaegfadgaieffe 
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hcgacfeccfajjeibeeiggegggabceiagahiccjj 
#dbfbbjfifhhbjeccedigidcgfgfahdhdfcdbcgi
dhbffhhaffgbhghjhbhhghbgfgbdiafhjgjfdbe 
#fchceiiijdheccfgdceeeaiehiejegcggbiijbf
ajejbgbfbafhcejfjfbcjgddceabcgadiafegfa 
#dahdbbhgfdgehhefhgchgfagjffgdddieeifjef
gfjjjbicdcijbiigbeffejficciffejfcfejcfg 
#feieceaghchhhfibajgaegihbbebhibccecjfjb
hfbjcdjfigideghcgefbaaiccjaifhjcdhddicj 

#ifhdideihfhbcifjdjgdgdaihjgajjeeggjjcei
iijghejdbfdeicihbjbidacedjbgjffcbgidfdj 
#ebhfjidfcijjjbjcicefiedfefdjibcfjhagcgb
hcgbaiciicidfehgbjhghhifjeigbhaeafjehch 
#bchfgfedaffieabchjgdcefigehcehgjahehcgg
cgaaagbeihhhghjecfieghiedbahjfibihghaif 
#iiaabcbfjgjdjabgdgifhhhhejchehicehhahdg
ciidcgaaahhceggaahgigccebacffceaijjfcfc 
#cefccfabeejeiaigdjiaihbjjcgeaajjbcghijf
egdfieiebcbdhfceecicfhgaibghdfegfbjhehf 
#feiageeaaeiddgaehjgfejhieigjeabagfebfjc
eejcjiijdhfhdedbdieegifijcghaciefefceid 
#dhbiigdicccdhigbicbhbfccdagbfgibfeafffc
ajiigceegieidjbhgcagadcifdchdchidbffaja 
#geabfhdhgibgbafcjfcijjdbdbdffibggigddfe
gaddgfghhjbfgabjcidhbifbagcbhjjgjjdgcjh 
#igdbdcffigfdfcefeghagfcfjeaiigejbdehfdh
haiaighacfeeidcjiebcgahebdcgjhgjaaejibe 
#fajccfdafjadbbdighhfaffijhjhefbgedgeidf
aacfjaicgghcjedgdgiabihiijgddbgefedfjgj 
#ghciacjaahhehhhbajaefiedcdcffdjcbcjchab
jebbbjibhcbefabcbfhdichabgegibcdjgfjbee 
#efaghgcebeifccgeffdbgiffffeifhacagdaibi
iajjbbbjhceiafidcefhdccfcfhicjefegghide 
#aefifidgfjcfjgagfjdccgabfgjbhcgagcgcaia

caigbhegejddfahjiheajhcajheiagdajibidia 
#cfajjfcafcjfcahjcgaiigihdbfiagidjhahbfg
hgdfbdbehbcdedffifdhiebbfgjdjgeifegjfjf 
#jgdgedhcidfabjaafgdbghadigjgbcdfjihbhca
edabcdidadbifefefeabajfgjeffjjbedhghghc 
#ehibcgefeacjdajghajeiiaaghbgfiafgjeiach
dcjdcgeaigiaifjhgdedaibhjeejjehfibhijdi 
#biagaaeeffjciicahiabfhdgebcabaedfjddebc
gaiigajabhcgfiejhgahhafjbjfgefedgbbhicf 
#jjegfeefcbejaecgjjbbdfbfcgbfjagcghachah
eaibhccaacbbiddiehiggbedaagbgjebedfggae 
#cffiadbcdaedddedgfabhibcdhjbddhbjfhdggb
ahidcdjbdjgbagjebihhbccehadeegbfadfebgf 
#hejfbhdibhcadaecaejcdhdgjgfgdjchgcfhcae
gieiejjeaebidaaaiaedjjfiaicgijbhbbgdigd 
#eedgdfjcbacfffhfdbacgihgecgiedegbfddaee
adbagehadciebeeceagddfjjghiejchggadhebe 
#ijadfchcieiedcgeifbbejhciiabhgjgfcefhaj
bidbfgfgdfddeiggehjgjhbeihbcfeiaefgfjjh 
#hfdfcaejadaejijjijbbbiejiefacbidicaeaga
fbgcgccdigbbhjcghajjeecjajggfiaafgdefdj 
#hddcfaeefcigfcdabihccaifdaaiajhdbijjhbd
afbbhcfiiihafhjbehhfjegeaadgfjbaeddafdj 
#acdfghbcgjciagjcfeifacajhgjcffaefdfaabj
ifbejabfadagdbfjdbdaechidebjaafbbegdhcb 

#begddcbiehbaeedbhfcdfidadaahighchbbcfda
ieafcbfbadfiicgiehcjjbahbjgdbgdhaigedhf 
#cfgjjgbdcdbdgaefbggaefchdddhijjiacbfdbf
hcieegdegbhjchchggihebfhegfbhfigjdiggee 
#hgebfgegabcabgcegbdfiahebedcjgafbjifcee
addjcigcgecibehedehfdgjagcafbccdechaeec 
#gjbfjiggdfjjcgbhffbbhgjfbhfdgijbfbfgcbh
cfeifgceefbacijgjaadbgaebheifhgfeddjdhj 
#daciiieaidbccbfbieebhgdgaajajdhhahfacac
icgbehcfhgaaehfaiiachdbcedbjjecdjiagidj 
#jbijbifjhdbheadebacihiaiejhfdjcfdaiibed
ifcccfhfcbjcbdhjjhdbbhiaedgahjfdaiccjjf 
#hjibgaefcdceebhjbgcaccgjhegggfdhiigaaid
cfiihaiciaaefjdjcgaccfbjaeddhifhchfhhfe 
#hiicageidhfbhbfgg                      
----------------------------------------
-------------------------------------   
,! ,smalle/ ,pisot-,vijay>aghavan numb]4
                                        
#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;inf9ity72 ?is
 numb] is also                          
#a_/b                                   
co%7#a_/b #c    ,pi7                    
-----------------                       
,pi to #bjjj digits x is444             
#b4dbhahigibjihhgjcbhgcfjdadcfagiadfceha
ahcfbiddghccijdigfcbgdiigdgbfdddgcdabjh 
#fhcfhabchjeejaegbjeijdchhachjfhjacgggje
hgbiefhdgehiifficfjcchcfahgcbdajecjjcib 
#hfehjhffibddcidfeghijhibdiafeacgajffcgg
ajdbebcbcfgcfjgfgeadgigifdffhhfcihcfijd 

#hejcicgciccbbebadehjefdjcdgbefeibedjbij
jbhhcjhfgjjaigicihjgdbachgbebbgijjhicbc 
#aedbbhejbbcfeaeebahhehaeaidbigefjhibcfc
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 ! ,ramanujan      
numb]1  to a precision ( #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 
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djcdhighghaddcacbbfgidajhcieaicfjbfejbh 
#ifjgbfecgbiabgfbbfichbdbgageeabghbgifec
gejgjjghdjjaaijjaibdagaccehcbgacdgjaeah 
#gefiebcahiejeggebbhifadifhbhbafejghbaff
heefjebahfbbbhcgfaeaajdebijgjdfeaihacej 
#fbdjfdjaefiieeejeefjggbcebgbcehihceibfg
iichbjijecbdahdjehiabgddhjadcidgdegjiej 
#fdgehfeeeaidgefjffcdgajgighcfffabibgfdg
ibjijifhgijcacahfeeedbhbgcbjfbfjfeicbdh 
#dacbfaebcgjehijjihbgecgjgaecgcfcjggbehj
habgeehbfibjhgbeiaehaijbjjejcigeaaibgbf 
#bhadbjeaebiehdhbhdfbhfjdhdjgadhjfgdiicc
gefhigedhafihigiaaffabejcbjgchciifcbidg 
#aigdgejffjhjgdciabbhbbeafajbihgaecabaec
ibhfgcbhijefdeeafheaajideajhejbdahfhhac 
#ceggecichcaiihhgeacafbegcddgiiidaajhaah
gdjjifggjfhbeggdejiejeibgieaggijjecdbbi 
#bbgfbeaceaegfgaciccebeecejhfihaicfdiech
aecchhbcihgjgeifgigfdgfhbejiacddbdbgbaa 

#ecgefbidfjicegbghjjbhjgdeaahhigcehddcab
beiidjgcehai                            
----------------------------------------
-------------------------------------   
,! ,robb9s con/ant4  ref3 ,d4 ,robb9s1  
,av]age di/ance betwe5 two po9ts 9 a    
box1 ,am]4 ,ma!matical ,mon?ly1 #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 ,con/ant  ref 3 ,d4,h4 ,lehm] 1  
factorization ( c]ta9 cyclotomic        
functions1 ,annals ( ,ma?1 ,s]ie #b1 vol
 #cd1 #aicc1 pp4 #dfa-dgi4  ,david      
,boyd1 ,small ,salem numb]s1 ,duke ,ma? 

,j\rnal1 vol #dd1 #aigg1 pp4 #cae-cbh   
?is numb] is an algebraic numb] ( ! #aj?
 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               
----------------------------------------
-------------------------------------   
s97#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_/;b9omial7#b99n1n71n#a44;inf9ity7
 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_/7n99b9omial7#b99n1n7711n#a44;inf
9ity72 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;inf9ity72 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               
----------------------------------------
-------------------------------------   
,! ,travel+ ,salesman ,con/ant1         
conjectur$ 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              
----------------------------------------
-------------------------------------   
,! ,tribonacci con/ant1 is su* t        

#a_/7a-x-x#b-x#c7 once exp&$ into a     
s]ies w give coeffici5ts proportional to
 approx4 c9999n & 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             _/    9 fact ! n'?      
,tribonacci numb] is giv5 by ?is ,,exact
 =mula4                                 
----------------------------------------
-----------------------------  ,see 3   
http3_/_/www4labri4u-bordeaux4fr_/loeb_/
book_/#ibpl4html            ,comm5t     
calcul] le nieme nombre de ,tribonacci  
,resume ( a conf]5ce giv5 9 #aiic       
7,univ]site ,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 ! actual n'? ,tribonacci numb] j    
r\nd ! result to ! ne>e/ integ]4  ,"h is
 ! =mula 'lpr9t$'444                    
#c997a_/c997aic99cc7a_/b777a_/c7a_/c997a
i-c99cc7a_/b777a_/c7a_/c7n_/77#ehfajb99c
c7a                                     
_/#b777b_/c7d-b997ehfajb99cc7a_/b777a_/c
77997ehfajb99cc7a_/b777a_/c72  ,?is     
=mula has #b p>ts1 fir/ ! num]ator is ! 
root ( 7x#c-x#b-x-#a7 no surprise "h1 b 
! d5om9ator 0 obta9$ us+ ,,lll          
7,p>i-,gp7 algori?m4 ,! ?+ is1 if y try 
to get a clos$ =mula by do+ ! ,z-trans=m
 or any?+ classical1 x won't "w v well  
s9ce ! actual symbolic expression w be  
huge & won't simplify4  ,! num]ical     
values ( ,tribonacci numb]s are c9999n  
ess5tially & ! c "h is "o ( ! roots (   
7x#c-x#b-x-#a71 !n !re is ano!r con/ant 
c#b4 ,s ! exact =mula is c9999n_/c#b4   

,ano!r way ( do+ 'exact =mulas' are giv5
 by us+   function ! n'? t]m ( ! s]ies  
expansion ( #a_/7axx9999#b7 is  #a - #b 
floor7#a_/c n  #b_/c7  floor7#a_/c n    
#a_/c7  floor7#a_/c n7                  
----------------------------------------
-------------------------------------   
,! tw9 primes con/ant4                  
#j4ffjafahaehdfhfiegcibghabaajjadeeegghd
cbfbc                                   
----------------------------------------
-------------------------------------   
,! ,v>ga con/ant1 also kn[n to be !     
#a_/7"o-n9? con/ant74                   
#i4bhijbediaibjhahiahgeeddidceieagdejfaj
cag   ,"o-n9? con/ant is                
#j4ajgfeciaibbfdhdegffaecbcddejijidgaije
hgigfejch                               
----------------------------------------
-------------------------------------   
#j4dgdidicgiihgibjfejccbejdfcfcbgihbifhe
eiedicgcbagbjbihbbhcccajbdh             
#fdeegibiagdhhchfjbgdbgefdabejejbadddahi
jcghdidbfbciedfdggebejdee               
#bjiigghebciejhhbghjhadhbaeibjhbefebjbia

baicjdaggjbhaieiihggihghg               
#fdjdcdbchjcecagiagjfbejafagjbebhjchdaee
cfhaigefgiahidhieibjhcheh   to #bef     
digits is also ?is clos$ 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 
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heaffbbeifeiijjdghhfaijffjgcjgcaegaajaf 
#ieheddgdgihghfichdcihcdjdjdaaaedfcbijgg
faijgidfdfgaaahfgcbdgafjcfbbaccdejjghhd 
#hbijdjfaafbgdcibgfcjbicicdaiiiehcgdcajh
acfecbifhidbfegahbeciecjhgcieadejaiahbj 
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fhefjjcbbifchdbdegihedihcagijgabfjiajaa 
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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 ,ap]y con/ant 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 
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idcjfcgjiddedehgjcjgehdahgeiffcaehhjdgj 
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icbbdcfbibabicgfhjbadggfbgfihhbbgbgeafa 
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cahefbhjgebbfcdgabgeffhcbjegcbfgfhgcjdc 
#fijhgbfeebfgdddiajgcjhhgdgadcdggagfhhac
bcheijebgdafhcjfacggcbjgdjhgfagecdeiifh 
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eebhdbdhiheceedhbbbheicijagbjffgafjbhde 
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dhebediafcedibgeeajdibfedfefjbfiheefcae 
#cciidieghhiefjfgghcchahfejjadfifebbcbcj
eccecdffiiebcfiigfcdccgifgjdbbhgfdifddf 
#iefgedjdifebejciiggjfjghdcdfcaibieefchb

hgagcghfjjdgejdiggjbaefgfdbiadafcbjhcee 
#acdhjegibeffjfhbehjbiidfaigafaafffbfbif
bjfhgcihecfecccahbhggbceifefdejbeeeiacf 
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#ebigfbfcdjgfaehdcaahhheeiicbeigiaicbgac
ebjjajbgdjbjgiahaagehgijgjgdgidafedgdhi 
#ajfahcfjgfagggcfghedgcdiaacjgbiafbhagfb
ijhhifcgbhggbjecfbagihecdhfhhjagdfiidgj 
#ajgchcgieicdbjaecechaibdccbicbfjaiiegdb
cgaadiddeeedacgfddfdihcchgdgccfjjiagbhf 
#icijedbgebehifdifibbacgcdchaeifhaeacebg
cfeaebcfhafjheicdgddicadcdhjgeijjcdebdh 
#jbbcgabdhcdjgjihicifdedaaechffadbaaiibg
ifehcjadcdeigjhjjeegefdjbaifjabacffchjf 
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hjibggdfacdeifebdafigecjdefghaeciaachhh 
#ebahbaceidhfjihhbbhhdigaaigdhhdaiebhddf
ciajgediejffcecbebbfcigicifbefghjefhjbc 
#dabiidcfajbjhhbecjdfjcaecfifaejajfaeijg
fdjchhddijjbhgefdiaaaiefgfajjdhicdddghh 
#bfbffgfcfhiadhbhfhgihceedabhfjhfhjhchha
fggbehficfebijdjjgdigieeafgbacieijhhdcg 
#jadbgeaagfidabihedbgjjejggaidjjbggjdhee
cfhebgddbdgffjijehifcjdgceeecggbfigeceb 

#fjfhegdhcbgfgjdjfcidiaihdgdhcgfjaehdhed
ahcdegfbbfdcfhhfiajfhbeeahbhebfacjgbdhj 
#bhjaafdhcfcigeeiefafihfihcicbhiccdhegea
bahegeaedihjccejfijdfcgfcfajjideegbahgh 
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ceeabidgifahcihffjhjiijbaijeibgcgfjafbj 
#afijdjhijhajfdfejhhihdfjchjcagedcgbaecd
jedafijfejabdddcagajeacagbaihfcjiagbcgh 
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fcacjcfbjejhdbdbffibcbejdgahdffeibhhaej 
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ddchfiaeidafbhbdhafdcgghjedehchcfhbhjah 
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bjabbjiiggeffcggebiddbgbibcjcijejhdjehh 
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bgfadhieicahbjdabcgaeddbegbbfaijejfdjea 
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ibdhjgiigiabdgcjigiidaieeaiejbibjjebhdc 
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aijabjdgagjcgcfhicifiafbdhagdagbiifbigj 
#fgjjecaibddhjafccafdefddejaccccijgfijdh
ibehbefbjdjdijcadcfhiiheiajffgdeebcaddb 
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jgccbhjijjfjjgggcdhjhabfhdjjefjgaeahfag 
#cefffhajghbajcebcjchbbhjidghbjaeccihgfb

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ijhjedjjddcgajifbhagiabdhhgbhdjfecbfdaa 
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#iacgaiffaegjjheehbbdgbcicibeddijhhhhcjj
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ijaegcecchddfgbjafbhghfjihcfgcddefeiefc 
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ajfjdjfajcijjidfefdchcbjijcjiiahgigjejg 
#cgihhadebcbgbehhghahihccchcbjdebebdjjcg
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ihcjibiadjhhgbfdibaahdigdcbbffcgabbfhbd 
#hgeidhchifadihagafhefgebcghejafjfhdecbd
ceidjdggfaeadihiggiddgfcbhhjhgdijahidca 

#efgihjhgaihddfgeggcdefheabhbaaiajfhidcf
hfaicaigcdfgddbiggijjaiafhfajegjiifjgjd 
#bdgacifdjjbdiahgffhfafgaieefahehaiagdgc
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ihicgajafjjhcjijfjaijiibdabhjijhhfaddda 
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fggcadjedgjhihebjhebjfaadceecedjhjaffed 
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fhjacjjbjghhhghbciedjijafbffiegfbbbjbje 
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edfcjjfdaafdghgdjddjfhdfjjjdajggibegfed 
#effjffaiicdabiicbgafhidfhcddefhaghhgggf

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#cjihfdcfejffihjdhichcejiiafbhfjbbdjifac
iajjggefhebfajijjihjchjafigefcbbedajhha 
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hgjfiadcdajffdhidhgfabcgcfdfhfeefdibbfg 
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dffhjbgfbjgdabdfdheahjcafcaehjbjheefgga 
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bhjheheggcejicgghefjecgiecdjfediheeadig 
#hffigbdabaadccicdjedgabccjiaefjiibcebbc
aehigfeiiedchfhfdfeedbhcaafhifjeeeccjbf 
#ehhhhddbdagcbjfbeedhijehajhcdejgfaiaige
aaahechcdicfbaafbchchhdeehfhdfcbdceehga 
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ajgiififjjbiagdjbcajgdijeebaiaaefjcdehj 
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aicgfaibegdcfjdjhdgiaececgbfcecfafgabcc 
#gjegfebcabheeiifbahghifgggcibiaieghjada
ccgejbchbcbhehfjdaddibjehecibfachfiieaa 
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#cgahidjihebagbediiajaeicagfbidaagcafgeh
egfjicehcbhdihaejdicfbhhadgigdadcgigeej 

#bdefegdaaecijiihagccijacagfbaegidbbfaed
hahafabggdaeibfhediacbiiegebjajfahgffdb 
#hfbabagighjfghjihcfbhcaefhbacacfehfhcgg
cgdhbhhejabiifebjcgabifbjefgiehbeeiigji 
#fdfihacjdhibecjejcgbhabgaiahjhaeafgbacb
ffjedfgcjhacbjaebgagggeicfecigcefjdheaf 
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jaggibhbidgdjbiibcfebjaigghiicgchiadiba 
#aagdgjjcidijhhbgfajiedhaggbcbjeahddefbg
aejffecffhfbghcbddiiiijdjedjaggbiegbcij 
#gcbechfjgdcbahadhiiccjciheaebhidcffifdc
ggcebhigfccbdgdagafjhfcjidajijheieigcfc 
#aadghcahhcabbcigghfjiiibigbeageifccgghh
iheaabjhdfbccdiejhbhjhdcbhabjjdbhjehfjg 
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eifjdhddadgegbhhfiejhedjefgcafgfhaadcdh 
#hhiachifaebjcjjghhicdfdbcjibeieihgaebgc
jccghcbiaecddfegdfhidgcgddiecbjgieiicjc 
#fahbffcfhjcacbdbcjggdjechibddcfbcjhfbah
cghbbhabfdciiehifhdbdddehigbjjcfadeicfd 
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iiggicahibhgiejjbbfhaajecfjcgbaidhciecc 
#jafagfdjchehdhdeedefdajhejgjfeaccabadhj

cdeajecbddchjiijdjcahaijieehdiaigdigjfa 
#ijdgdeebfcbhejibeeibhefjcbdgihihhejhdeb
jjhbedfeidehehjgeaifeegdjgiegeffdagbaea 
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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 !                            
sum7#a_/n9999#e1n#a44;inf9ity7 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;inf9ity72         
#a4jjhcdibggchaibbhbfhcigigedihdigifgeie
iihfcefjefebchgjfdagbhcacf              
#egafjadghcagceegcecdfjififhiacheacbcifh
ifadecfeadiajgdhhgbhfgggd               
#aihdjcceddjcaegihcjajccihdefbabajfidfce
hebdcijfehccecifdfgfiigef               
#gfiffiadbghjdcadcccidgdiebaecghijbhjjbe
ijdeeeaigicecajhcgjjhdbaj               
#gcbiciijdfajgjhefdabcefjehijfbbeiiggfji
hfidgedjgfcbjjjjdhafcbiea               
#behfgfibejfcjgcddacfcbeeefjacfjcjejjgcg
ccjbdacahgjcgieajbffbdggi               
#ciedfejbbedfgjdbjaeeajdjeehbbbdbcibejea
jhfhhcggbgjggdbfjjbaggajj               
#jaiedeegghihihcfjdfgdedjfabaiebfejgfedf
aafacefedhfgiaejjhjheheed               
----------------------------------------

-------------------------------------   
,zeta7#i7 or                            
sum7#a_/n9999#i1n#a44;inf9ity72         
#a4jjbjjhcibhbfjhbbaddaghebgfibcbdabjfjd
hefjeheacidhhhgefedheiffae              
#ijighejeccijbehcihiejcicjfiabgafiehfaeg
djhfjdgfehdgjfjbfadbecgci               
#gjgbbdcjaecjfiacbdihgfdbeajijibidhfhgfg
fedecifigidaedjghbfjbbifd               
#aeddhcfbejffhfbijefgjgcfdebafjaecadbddb
acbfccgeihhaeeehjebeiaded               
#jhdhijaeciebggdgdefaccdeajbhgdjfacbgdff
jfibgfccijjafbidbgjhfdbbj               
#aabcafbbjibdabfegeccbfbjedfbbicbaededff
eagiidejchffbgghbbcefdggf               
#affjccjbhadibcfdegjciicjaaaichcihejagaf
gibfjjbjfdibcjfigiehejide               
#hdegiffedhedjjbfideaahgeidhaefadcjcgegg
faedddccdccihciiheadaichc               
----------------------------------------
-------------------------------------   
,?is numb]1 ! ,product,cos,pi_/n1       
n1#c1;inf9ity  is ! limit ( an int]e/+  
figure 9 geometry43 ,if we take a       
circle1 inscribe a triangle1 !n incribe 

ano!r circle inside ! triangle1 !n      
inscribe a squ>e inside ! inn] circle1  
!n inscribe ano!r circle inside ! squ>e1
 !n inscribe a p5tagon444  ,! radius (  
?is figure 7! numb] ( sides ( ! polygon 
increase ) e /ep3triangle #c1 squ>e #d1 
p5tagon #e1 4447 approa*es a limit3     
,product,cos,pi_/n1 n1#c1;inf9ity ,is   
!re any way to get an analytic solution 
to ?is8 ,l ?is w\ld be ! squ>e root (   
,pi or "s comb9ation ( radicals &       
irrational numb]s8 ,anyway1 ,?anks1     
,m\nitra ,*att]ji m\nitraseas4ucla4edu  
m5tion$ 9 decemb] #aiie4 ,by  ,m\nitra  
,*att]ji                                
4#aadidbjddhecbifbjjgjajdjaegdfieihgdbhc
jgieccgbjjhfceafhddjbccife2   maple     
r\t9e --                                
product7cos7,pi_/n71n#c44;inf9ity72evalf
71#fd72                                 
----------------------------------------
--------------------------  ,! reque/ 0 
s5t by a*im flamm5kamp on ,tue ,feb #bg 
#ji3je3ac ,,p/ #aiif ,! email address   
is3 a*imma!matik4uni-j5a4de ,! numb] is 

#a4fjadjbbdcedihhgfaciccbe  7to #bd     
digits ( precision74                    
-int7sqrt7x7_/log7#a-x71x#j44a72        
----------------------------------------
---------------------------             
4#bhcbfeabacajcjggcbehgfheedjdejhehhfhde
babcjgeiacdgidiefjicjcbddgfjbhibjgdffgjc
eeabjjgbhcdcbdfgahbff                   
#agbagidgjfcbfhgbchibcgdahbfebgcaifchiaa
fibiabahaigejhhhjfbdiebidbggbefaiagaidbd
bgcifgchdedeijhajffaf                   
#eabdgjbcbbeaceihdacchhibjbacchgececejfi
bcfbhffgjggehcgfachhehdhbbffibhccbgahhhb
ahfdgchiabebdgjfbfaic                   
#aacdafbjgedjcjjhjcghhadiifaebdjfehaejic
fffagabgedhgdebiabjgfibgijghhbfadfibejfi
cciaehhbdcggbejghjjjf                   
#hafiafhcfehdccechdhjecceahjdcadfdjejcjg
eddefbideggigeeehaggadbddghgbefbhbiaeg  
,!re is a patt]n 9 ! b9>y expansion (   
?is numb]4  ,! reque/ 0 s5t by ,b4,j4   
,m>es on ,sun ,dec  #c #ae3bj3ah ,,p/   
#aiie ,! email address is3              
bjm>esteleport4com                      
----------------------------------------

---------------  ,! reque/ 0 s5t by ,joe
 ,keane on ,sun ,sep #aj #je3jb3bf ,,pdt
 #aiie ,! email address is3             
jgknetcom4com  ,! numb] to be te/$ is3  
#a4chdcbifiafefghfiafcffjjjgjdfiahgbgeii
cfjbhidfgbbhjjcafhbhfchghjfijhhbajhjhcef
cde  ,! numb] ( correct digits 9 !      
numb]3 #gi  ,! h9ts giv5 by ! us]3  ,x's
 log77#csqrt7#g77_/sqrt7#b77 or         
#a_/b99;arcco%7#h74                     
----------------------------------------
----------------  ,! reque/ 0 s5t by    
7,mr47 ,b4,j4 ,m>es on ,sat ,dec  #i    
#ai3aj3bg ,,p/ #aiie ,! email address   
is3 bjm>esteleport4com  ,! numb] to be  
te/$ is3                                
4#hfbbdjabehfhjedegaeeggijbhcbdicideghef
egfdgdbgfhbiijideafjgabadeegcjfgdjeijeaf
dehjdbjchddadchfahac                    
#deabegbbijcjccjieheacijhaaadijijdcgbgje
fcaijdhcfgiieagccdfjiiceeffhfdbjcehaiaaa
iihgggbeifiebhhhcbdc  ,ano!r b9>y       
patt]n4                                 
----------------------------------------
-----------------  ,! reque/ 0 s5t by   

,jon ,borwe9 on ,sun ,nov  #e #jf3ji3bh 
,,gmt #aiie ,! email address is3        
jborwe9cecm4sfu4ca  ,! numb] to be te/$ 
is3 4#jaaahfhjjjcbhggajghgjjdfha  ,!    
numb] ( correct digits 9 ! numb]3 #bj   
,! te/7s7 to be p]=m$ on ! numb]3       
algebraic                               
----------------------------------------
----------------  #a4defgiajcajdfijg  ,!
 numb] ( correct digits 9 ! numb]3 #af  
,! te/7s7 to be p]=m$ on ! numb]3       
algebraic gammamultiplicative           
gammaadditve zetamultiplicative         
zetaadditive psidigamma                 
l9e>dep5d5cesalvage  ,! h9ts giv5 by !  
us]3  p7#j7a q7#j7b                     
p7i#a7sqrt7p7i799q7i77  i  #j1a1b144    
q7i#a77p7i7  q7i77_/#b  i  #j1a1b144  x 
lim   p7i7    lim   q7i7     i-inf      
i-inf                                   
----------------------------------------
----------------  ,! reque/ 0 s5t by    
,olivi] ,g]>d on ,mon ,jan #bi #ah3dh3db
 ,,p/ #aiif ,! email address is3        
quadratureonco4te*l9k4fr  ,! numb] to be

 te/$ is3 #a4jfbeejhjedifbeeich  ,?is   
numb] arises 9 ! /udy ( g5]aliz$ ,zeta  
functions on non associative sets4      
----------------------------------------
----------------  ,! reque/ 0 s5t by    
,mi*ael ,moss+h(f  on ,fri ,feb  #i     
#ad3dj3bh ,,p/ #aiif ,! email address   
is3 mjmma?4app/ate4edu  ,! numb] to be  
te/$ is3 #a4bifbajfeieiccji   7see bel[ 
= #bejj digits ( x74  ,z ,i m5tion$ 9 ! 
orig9al note1 x w\ld be int]e/+ to see  
if ?is numb] satisfies a simple         
polynomial ( degree  #cd4  ,! simple/   
polynomial ,i "k ( t x satisfies is     
x#ch-x#cf-x#cd-x#bix#bh-x#bd-x#adx#aj-x#
i-x#d-x#ba  ,i f\nd ?is dur+ a se>* =   
polynomials ) height #a1 degree #ch1 &  
,mahl] measure  #a4c4  ,i also h a      
second new ,salem numb] t w\ld be       
int]e/+ to try4  ,?anks = runn+ ?is6    
,be/ reg>ds1  ,mike ,moss+h(f           
mjmma?4app/ate4edu                      
#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                                   
----------------------------------------
---------------------  ,ref]5ce         
,philippe ,flajolet & ,&rew ,odlyzko 9  
,r&om ,mapp+ ,/ati/ics y c h ! article  
at                                      
ftp3_/_/netlib4att4com_/netlib_/att_/ma?
_/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             
----------------------------------------
-------------------------------------   
,! ,h>d hexagons ,entropy ,con/ant   ,! 
h>d-hexagons entropy con/ant is         
algebraic 7see bel[ z numb]74  ,! value 
is 3                                    
#a4ciedheigbdgicjbgcebbiejjffceffhhhjfhi
edajcgbhaddffaaijhagdgbaefacegfjhhjcehf 
#igggdfhihcghgcjhebgedbgijbffhifhefjgfhe
fegahdhdbbabdegaaieaafcicdihahbffidgjhc 
#bebedgagcgididgecdhfbbhabbiabfahgbhaeed
cdjabfafbgdgcefigcehegjihbcgefhabhihdad 
#idhhjjjaficdijcgbciiefebjidefhbecegbech
fccegbjjebaaibejgdgcihaajaeachjhfbhiffa 
#bfhacfghghcahhefcjdjdfhbgdgajgdggbjdfhf
hidgeffegehjijeecjbgjjfffgedjdifbdbggai 
#jfjhedecfadbbafhcfbificcjafijjccjicgbgf
ieffbabficihgbfhbcajdibcjdgddbhhbeadgha 
#gjbiffajgbgjjedbibhabbhjgiejfaccfcbaeej
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#dijhidbdicbicjgedjhafbajeeecbhjfhfajfai
gjeedejcgieejggehjgbeecgfachehjccfaieje 
#bajiehifggbifiidaffcjidbfjafaeeffibebah

ediccfdgfifheeahbdbhahidfaejibheefdigdh 
#ejaceiijfibiaebegahccheajhjbabhaajdigee
ccihdgcffibgiadcihhibjdaheaceehcacjcege 
#fgcgajdfebbdhjgdedgddihbehchheahcbhgafg
cegadfjibjijgdcdjbheagdfegaefediijhbbib 
#iiihhdfabiifacgdgiiebcehccfejghfjggjeaf
jhghgifcabjbgchcejajbhieifehhajgfhbbddj 
#adfhabadgbfghijcehhhjjhheahaijecgdbhfff
fjeebggegbbgcdajecacbbeccg  ,tak5 f ,!  
,favorite ma!matical con/ants ( ,/ev5   
,f9*1 ,ma?s(t ,inc4  ,! con/ant is giv5 
by ?is 7see z bel[7444                  
#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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