The board / number theory / #1059
Erdős Problem #1059
OPENdatabase updated 2025-09-28
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 \(p - k!\) is composite for each \(k\) such that \(1 ≤ k! < p\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/1059.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA064152
Original pageerdosproblems.com/1059
Attempts
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proved the sharp bound \(p-k!\geq\operatorname{nextprime}(k)^2\), certified equality solutions \(769\) and \(6\,227\,021\,089\), and exactly corrected/enumerated the strict sequence through \(10^9\) (
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