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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.
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Greatest common divisor of a^{2^n}-1 and b^{2^n}-1

Let a and b be 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.