erdős #1065
Are there infinitely many primes such that for some prime and ? Or ?
Worked, still open.
number theory · open · formalized (Lean) · 0 attempts
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vela registry pull vfr_37aec80d874a0239vela reproduce examples/erdos-problemsevidence
unverified AI candidates (2)
gpt-erdos · GPT-5.2 Pro + Deep Research · unverified
For $k=0$, (p=2^{0}q+1=q+1) is even for every odd prime $q$, so the only case is (q=2\Rightarrow p=3). So the real question is for (k\ge 1).
candidate solution ↗llm-hunter · gpt pro 5.2 · unverified
1 LLM attack(s) recorded (gpt pro 5.2); unverified.
candidate solution ↗formal
AMS 11 · open (literature)
theorem erdos_1065.parts.i :
answer(sorry) ↔ Set.Infinite {p | ∃ q k, p.Prime ∧ q.Prime ∧ p = 2^k * q + 1}formal-conjectures/1065.lean ↗oeis
A074781 — Primes of the form p*2^k + 1 for any k and any prime p.3,5,7,11,13,17,23,29,41,47,53,59,83,89,97,107,113,137,149,167,173,179,193,227,233,257,263,269,293,317,347,353,359,383,38A339465 — Primes p such that (p-1)/gpf(p-1) = 2^q * 3^r with q, r >= 1, where gpf(m) is the greatest prime factor of m, A006530.19,31,37,43,61,67,73,79,103,109,127,139,157,163,181,199,223,229,241,271,277,283,307,313,337,349,367,373,379,397,409,433,
status
open