---

Claim. Every infinite sum-free sequence $A=\{a_1<a_2<\cdots\}\subseteq\mathbb N$, in the sense of Problem 876 (no term is a sum of two or more distinct smaller terms), satisfies

$$\limsup_{n\to\infty}\frac{a_{n+1}-a_n}{n}=\infty .$$

In particular $a_{n+1}-a_n<n$ fails for infinitely many $n$, so the second question of the problem has the answer no, whether it is read for all $n$ or for all large $n$; no bounded multiple of $n$ bounds the gaps either. The claim was submitted to the site's proof-claims tab on 2026-07-18 by Liam Price, who credits the proof to GPT 5.6 Sol Pro, and its summary says that the argument uses Corollary 1.2 of S. Fan, Strongly complete sets and a conjecture of Erdős, arXiv:2607.14071, the preprint the claim describes as a resolution of Problem 254 and which the problem page cites as [Fa26]; the preprint's library card is Fan 2026, with Corollary 1.2 (p. 4) claims checked. The write-up is a read-only document on a collaborative editor, linked above (read status: unread). A comment on the claim thread sketches the route: a sum-free sequence has density zero ($A(t)=o(t)$, Erdős 1962, Theorem I) and satisfies $\liminf A(t)/\sqrt t<\infty$, since otherwise its subset sums would contain long progressions and, by the completeness theorem of Burr and Erdős, the sequence would be complete; if $(a_{n+1}-a_n)/n$ were bounded, the tail greatest common divisors of the sequence would stabilize, and after removing finitely many terms and dividing by the common factor the sequence would satisfy the hypotheses of Fan's theorem and be complete, contradicting sum-freeness.

Submission note. Posted to erdosproblems.com as a proof claim by Liam Price (account Leeham) on 18 July 2026, giving "GPT 5.6 Sol Pro" as the AI used:

GPT-5.6 Sol Pro proves that every infinite sum-free sequence $A=\{a_1<a_2<\cdots\}\subseteq\mathbb{N}$ satisfies:

$$> \limsup_{n\to\infty}\frac{a_{n+1}-a_n}{n}=\infty.$$

Consequently,

$a_{n+1}-a_n<n$ cannot hold for all sufficiently large $n$, resolving the second question in the negative. Interestingly, the proof uses Corollary 1.2 from Steve Fan's recent resolution of problem 254.

Covers. The second question only, answered no in both of its readings. Not covered: the first question, how small the gaps can be (the claim gives no rate beyond the unbounded ratio), and the reciprocal-sum maximum asked in the site's commentary. The later full claim of Korsky asserts a characterization of the attainable gap sizes that would contain this statement.

Depends on. Fan's claim on Problem 254, through Corollary 1.2 of [Fa26], the preprint the claim invokes for Problem 254; that claim is claimed, not accepted, so the dependency is unaccepted. The route sketched on the thread also uses the density-zero theorem, Theorem I of Erdős 1962. The completeness theorem of Burr and Erdős and the subset-sum results invoked on the thread are literature results outside this wiki.

Standing. Claimed. The site's label is OPEN and its commentary does not mention the claim (page without a last-edited date,), so the curator records no acceptance. The claim thread carries seven comments: the author of [Fa26] writes that they checked the proof and believes it correct, notes that a lemma of the write-up was known to Erdős and that Theorem 3 of Łuczak and Schoen is a weaker form of one of its corollaries, and asks whether $\limsup(a_{n+1}-a_n)/(n\log n)=\infty$ holds, which they can prove under an extra divisibility hypothesis; another commenter calls the result an immediate corollary of Fan's theorem, with minor presentation issues the claimant agreed to fix. These are informal endorsements on a forum, not a review this corpus can list as evidence, and [Fa26] is itself an unrefereed preprint. No arXiv version or journal record of the write-up was found on 2026-10-07.