TITLE: Greatest common divisor of a^{2^n}-1 and b^{2^n}-1
QUESTION [11 upvotes]: Let a and b be coprime integers.  Do we know, expect, or unexpect that there are infinitely many primes p which divide
$gcd(a^{2^n} - 1, b^{2^n}-1)$
for some n?  Certainly any Fermat prime will divide both if I let n get large enough, but one doesn't know whether there are infinitely many of those.

REPLY [2 votes]: A comment on one of Joe's questions: Let $B$ be any real number. It is known unconditionally that there are infinitely many $m$ for which $\phi(m)$ is a square and for which the smallest prime factor of $m$ exceeds $B$. One can even take $m$ as a product of two primes here; see, e.g., article 4 from 
http://www.integers-ejcnt.org/vol11a.html
or an arXiv preprint of Tristan Freiberg.
If we choose $B$ larger than $|a|$ and $|b|$, then $m \mid \gcd(a^{\phi(m)}-1, b^{\phi(m)}-1)$, and so there is a prime $> B$ in the support of $\gcd(a^{n^2}-1, b^{n^2}-1)$.