Problem 1059

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claims/: The 1 claim page of Problem 1059, one per claimant's result; the problem's standing derives from them.

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Statement. Are there infinitely many primes $p$ such that $p-k!$ is composite for each $k$ such that $1\leq k!<p$?

Status. The site labels the problem OPEN, with the explanation that the question cannot be settled by a finite computation. The site's proof-claims tab carries two entries, neither accepted by the site. Johan Land's full claim, filed 2026-09-05 with AI assistance and followed by a Lean development in a comment of 6 September 2026, is recorded as pending on Land's claim page. The derived standing, claimed, proved, departs from the site's OPEN only by counting that pending claim, which neither the site nor an outside review has accepted; it becomes solved, proved if the claim is accepted. Aakash Gurung's partial claim, filed 2026-08-05, concerns only Erdős's easier auxiliary question on integers; it settles no instance of the prime question, so it has no claim page, and the Current assessment describes it.

Source. erdosproblems.com/1059, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1059, https://www.erdosproblems.com/1059.

References.

• [Gu04] Guy, Richard K., Unsolved problems in number theory, third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp.; A2 "Primes connected with factorials", printed p. 11: the question as stated here, with the examples $p=101$ and $p=211$ and the suggested easier variant on integers $n$ in $(i!,(i+1)!]$ with all prime factors above $i$ and every $n-k!$ ($1\le k\le i$) composite. Library home: guy_2004_unsolved_problems_number_theory.

Formalization. Statement in formal-conjectures. The claimant's Lean development for the full claim is linked from Land's claim page above; this corpus has not built it.

Current assessment

The site's formulation (accessed; label OPEN on 2026-10-06, with no comments, two proof claims, and the community database recording the problem as open) asks for infinitely many primes $p$ such that $p-k!$ is composite for every $k$ with $k!<p$. The site's label is the only source of its openness: no literature search beyond the site, its thread and Guy's Problem A2 is compiled here.

Two tab entries, neither accepted. Land's manuscript (5 September 2026) states that for any $K(X)=o(\log X)$, every large $X$ and any at most $K(X)$ distinct shifts in $[1,X]$, some prime $p\in(2X,3X]$ has every $p-h_i$ composite; with $X=m!$ and the shifts $i!$ this gives one such prime in $(2\cdot m!,3\cdot m!]$ for every large $m$, hence infinitely many. The theorem gives one prime per interval, so infinitude and not positive density. The claimant reports a Lean development, compiling with only propext, Classical.choice and Quot.sound, whose terminal theorem states that the problem's set of primes is infinite; its README says that no independent replication or external audit is recorded, the manuscript says that it is not peer reviewed, this corpus has not built the development, and no site acceptance, refereed version or outside review was found, so Land's claim stays claimed and the problem's standing is derived from it. Gurung's manuscript Factorial-difference composites with no small prime factors, filed on the site's proof-claims tab on 5 August 2026 and written with ChatGPT 5.6 Sol Max (Codex) and ChatGPT 5.6 Sol Pro, as the tab entry says, states as its Theorem 2.1 that for an absolute $\eta>0$ and every large $L$ at least $((L+1)!)^{\eta}/(100\log L)$ integers $n$ in $(2L!,(L+1)!]$ have every prime factor above $L$ and every $n-k!$ ($1\le k\le L$) composite; its sieve input is the published fundamental lemma of sieve theory (Koukoulopoulos 2019, Theorem 18.11(a)). The Lean development factorial-hypergraph-auxiliary-erdos formalizes a different manuscript of his, Factorial-residue hypergraphs and an auxiliary problem of Erdős, which reaches the same count through rainbow covers of a factorial-residue hypergraph. It takes the sieve lemma as an explicit hypothesis and is registered on the Palomar registry as entry PALOMAR-2026-08-27-000003 (27 August 2026), whose entry says that it does not solve the prime problem. The integers counted need not be prime, so the result settles no instance of the question and has no claim page.

The public-literature account of known results is not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• guy_2004_unsolved_problems_number_theory