Problem 1141

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claims/: The 1 claim page of Problem 1141, one per claimant's result; the problem's standing derives from them.

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Statement. Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2<n$?

Status. The site labels the problem DISPROVED (LEAN). It credits the finiteness theorem of [APSSV26b], whose authors attribute the proof to an internal OpenAI model; its Lean marker refers to third-party formalizations that this corpus has not built. The standing derives from [[problems/primes/E1141/claims/2026_04_08_alexeev_putterman_sawhney_sellke_valiant|the claim page]].

Source. erdosproblems.com/1141, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1141, https://www.erdosproblems.com/1141.

References.

• [APSSV26b] B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant, Short proofs in combinatorics, probability, and number theory II. arXiv:2604.06609 (2026).
• [Po17] Pollack, Paul, Bounds for the first several prime character nonresidues. Proc. Amer. Math. Soc. 145 (2017), 2815-2826.
• [Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).

Formalization. Statement in formal-conjectures, whose formal_proof attribute points to a third-party Lean proof; that proof, an earlier one posted in the site's thread and two earlier files in Boris Alexeev's repository, one of them avoiding Pollack's theorem, are formalization links on the claim page. The corpus has not built any of them.

Current assessment

The site's formulation (as of 2026-10-07, with its commentary last edited 9 April 2026) asks whether infinitely many $n$ have $n-k^2$ prime for every $k$ coprime to $n$ with $k^2<n$. The answer is no: Theorem 6.1 of [APSSV26b] proves, for each fixed $a\geq1$, that only finitely many $n$ have $n-ak^2$ prime for every $k\geq1$ with $(k,n)=1$ and $ak^2<n$, by a short deduction from Pollack's Theorem 1.3 [Po17] on small primes at which a quadratic character equals $1$; the case $a=1$ is the question. The bound is ineffective, since Pollack's theorem uses Siegel's theorem, and the thread records that an effective version would need progress on Siegel zeros. The authors attribute the proof to an internal OpenAI model. The accepted claim is recorded on [[problems/primes/E1141/claims/2026_04_08_alexeev_putterman_sawhney_sellke_valiant|the claim page]] with the curator's credit; the preprint is the source, with no journal publication known and the corpus has not built the four Lean files the page links.

Before the resolution, Tang, with ChatGPT-5.2 Thinking, showed by Montgomery's sieve that few integers have the property. Theorem 2.1 of his note A note on Problem #1141 (25 January 2026, with ChatGPT-5.2 Thinking named as co-author) states that the number $g(N)$ of such $n\leq N$ satisfies $g(N)\ll_\varepsilon N^{1/2+\varepsilon}$ for every $\varepsilon>0$. The site's commentary credits the bound, but a counting bound decides no instance of the question and the note is not refereed, so it has no claim page. The booklet [Va99] asks whether $968$ is the largest such $n$, which fails since $968-3^2=7\cdot137$; the integers with the property are OEIS sequence A214583, whose largest known member is $1722$, and [APSSV26b] remarks that computation suggests it is the largest. The thread also holds a summary of the argument for $a=1$, the exchange on effectivity, a Lean proof of Mertens' third theorem that removes one hypothesis of the first formalization, a forum user's report of 25 January 2026 that he and ChatGPT judged Tang's note correct at an intermediate level of scrutiny, which is not acceptance evidence, a sketch of 28 January 2026 of an upper-bound sieve that would give the negative answer under the generalized Riemann hypothesis and suggests that the exceptional $n$ are very thin unconditionally, and a remark that $30$ is the largest $n$ with $n-k$ never composite for every admissible $k$, by Bonse's inequality; none of these is a claim on the question, so none has a page.

Search scope (2026-10-07): the site page, its thread, the arXiv record and the linked repositories.

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.

• alexeev_2026_short_proofs_combinatorics_probability_number_theory
• alexeev_2026_short_proofs_combinatorics_probability_number_theory / theorem_6_1
• pollack_2017_bounds_first_several_prime_character_nonresidues
• pollack_2017_bounds_first_several_prime_character_nonresidues / theorem_1_3