arXiv:2611.00075v1 Announce Type: new Abstract: Let $\Phi_r(a,b)$ denote the homogeneous cyclotomic value. For sufficiently large $A$, uniformly for $1/2\leq B\leq A$, $r\geq3$ and $Z\geq2$, the number of coprime pairs in $(A,2A]\times(B,2B]$ whose greatest prime factor $P(\Phi_r(a,b))$ is at most $Z$ is $O(AB/\varphi(r)+\sqrt{r/\varphi(r)}\,\pi(Z;r,1))$, where $\pi(Z;r,1)$ counts primes at most $Z$ congruent to $1$ modulo $r$. If $n=n(A)\to\infty$ through positive integers and $1\leq H(A)=o(\log A)$, then $P(a^{2^n}+1)>2^nAH(A)$ for almost all $a\in(A,2A]$, and $P(a^{2^n}+b^{2^n})>2^nA^2H(A)$ for almost all coprime pairs in $(A,2A]^2$, with no further restriction on $n$. For every fixed $\alpha>0$ and every $r\geq A^\alpha$, pairwise distinct primitive prime divisors of $\Phi_r(a)$ can be selected for all but $O_\alpha(A^{1-c_\alpha})$ bases in $(A,2A]$, where $c_\alpha>0$. For coprime positive integers $a,b$ with $\max(a,b)\geq2$ and $r\geq3$, put $C=\Phi_r(a,b)$ and $S=\varphi(r)^2\log\max(a,b)$. Uniformly as $S\to\infty$, $\max_{p\mid C}p^{v_p(C)}\geq(5/16-o(1))S$. For every positive function $h(S)\to\infty$, $\sum_{p\mid C,\;p^{v_p(C)}>S/h(S)}v_p(C)\log p =(1-o(1))\log C$ uniformly in the same limit.
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