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Claim. If the abc conjecture holds then, with $s_1<s_2<\cdots$ the squarefree numbers, the moment sum $\sum_{s_{n+1}\le x}(s_{n+1}-s_n)^\alpha$ is asymptotic to $B(\alpha)\,x$ for every real $\alpha\ge0$, so the limit that Problem 145 asks about would exist for every $\alpha$. The source is Andrew Granville, ABC allows us to count squarefrees, Internat. Math. Res. Notices 1998, no. 19, 991--1009, whose abstract states that the paper deduces from abc that every interval of length $O(x^\varepsilon)$ around $x$ contains a squarefree number and gives the asymptotic formula, predicted by Erdős, for the average moments of the gaps between squarefree numbers; the site's commentary records the range as all $\alpha\ge0$. The claim is conditional: the abc conjecture asserts that for every $\varepsilon>0$ there are only finitely many coprime triples of positive integers $a+b=c$ with $c$ larger than the $(1+\varepsilon)$-th power of the product of the distinct primes dividing $abc$, and it is unproven, so this page derives nothing for the problem's standing. The gap bound $s_{n+1}-s_n\ll s_n^\varepsilon$ under abc is what controls the tail of the moment sum for every $\alpha$; without it the unconditional range stops at [[problems/integer_sequences/E0145/claims/2023_10_12_chan|Chan's $\alpha<3.75$]].

Acceptance. Refereed: International Mathematics Research Notices, volume 1998, issue 19, pp. 991--1009; the Crossref record of the DOI gives these data. The site labels the problem OPEN (page last edited 19 October 2025), so its curator's remark that the statement follows from abc by this paper is commentary on an open problem and not acceptance, and no reviewed evidence is listed. The corpus holds no card for the paper, has not checked the proof and awards no tier of its own.

Depends on. Nothing in this wiki; the claim rests on the cited paper and on the abc conjecture.