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Claim. Let $s_1<s_2<\cdots$ be the squarefree numbers and let $\alpha(h)$ be Mirsky's density of the $s_n$ with gap $s_{n+1}-s_n=h$. Theorem 1 of Tsz Ho Chan, On moments of gaps between consecutive square-free numbers, Mosc. J. Comb. Number Theory 12 (2023), no. 4, 287--295, states that for $0\le\gamma<3.75$

$$\sum_{x/2<s_{n+1}\le x}(s_{n+1}-s_n)^\gamma\sim\tfrac12B(\gamma)\,x, \qquad B(\gamma)=\sum_{h\ge1}h^\gamma\alpha(h),$$

and summing over dyadic ranges gives $\sum_{s_{n+1}\le x}(s_{n+1}-s_n)^\gamma\sim B(\gamma)\,x$, so the limit that Problem 145 asks about exists and equals $B(\gamma)$ for every $0\le\gamma<3.75$. The paper is on the card chan_2023_moments_gaps_between_consecutive_square_free. Its introduction records the earlier ranges: Erdős $0\le\gamma\le2$, Hooley $0\le\gamma\le3$, Filaseta $29/9$ and Filaseta and Trifonov $43/13$ (strict upper ends), Huxley $11/3$ and then $59/16$. The proof refines Huxley's geometric argument: a sharper count of rational points near a curve in Case 1(a) and a fifth-derivative estimate in place of a fourth-derivative one, with a finer split of the gap parameter, and the paper says that further progress depends on Huxley's Case 2. The site prints the range as $\alpha\le3.75$; the theorem's range is $0\le\gamma<3.75$, with the endpoint open.

Covers. Every exponent $0\le\alpha<3.75$, the widest range proved without hypothesis; it subsumes Erdős 1951, Hooley 1973 and Huxley 1997. Not covered: every $\alpha\ge3.75$, which Granville 1998 settles only under the abc conjecture.

Acceptance. Refereed: Moscow Journal of Combinatorics and Number Theory, volume 12, issue 4 (2023), pp. 287--295, published 8 December 2023; the Crossref record of the DOI gives these data, and the arXiv posting (v1, 12 October 2023, CC BY 4.0) carries the journal reference. The site labels the problem OPEN (page last edited 19 October 2025), so its curator's remark that Chan extended the range is commentary on an open problem and not acceptance, and no reviewed evidence is listed. The corpus has not reproved the theorem and awards no tier of its own.

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