The board / number theory / #1065
Erdős Problem #1065
OPENdatabase updated 2025-10-01
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Are there infinitely many primes \(p\) such that \(p = 2^k * q + 1\) for some prime \(q\) and \(k ≥ 0\)? This is mentioned as B46 in [Unsolved Problems in Number Theory](https://doi.org/10.1007/978-0-387-26677-0) by *Richard K. Guy*
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/1065.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1065
Attempts
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Proved the canonical reduction and a uniform \(O(x/\log^2x)\) upper bound modulo the standard Selberg upper sieve; exactly enumerated 481,051 first-form and 1,021,854 second-form primes through \(10^8
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