Problem
NT-B2-M03-P019 Infinitely Many Primes \(1\pmod{2^k}\)
#19
★★★★★ Level 5 of 5
Let \(k\) be a fixed positive integer. Prove that there are infinitely many primes \(q\equiv1\pmod{2^k}\).
Use prime divisors of \(2^{2^n}+1\) for \(n\ge k-1\).
Take \(F_n=2^{2^n}+1\) for \(n\ge k-1\). Every prime divisor \(q\) of \(F_n\) is odd and, by the Fermat-type lemma, satisfies \(q\equiv1\pmod{2^{n+1}}\), hence also \(q\equiv1\pmod{2^k}\). The numbers \(F_n\) are pairwise coprime, so their prime divisors for different \(n\) do not repeat. Therefore infinitely many such primes exist.
This is a strong challenge problem: it combines the two previous ideas into a constructive infinitude proof.