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Claim. Let $s_1<s_2<\cdots$ be the squarefree numbers. For every real $0\le\alpha\le2$ there is a constant $B(\alpha)>0$ with

$$\sum_{s_{n+1}\le x}(s_{n+1}-s_n)^\alpha=B(\alpha)\,x+o(x),$$

so the limit that Problem 145 asks about exists for every $\alpha$ in that range. The source is P. Erdős, Some problems and results in elementary number theory, Publ. Math. Debrecen 2 (1951), 103--109, on the card erdos_1951_problems_results_elementary_number_theory. The paper's moment asymptotic, display (23) on p. 107, is sketched for $\alpha=2$ on p. 109 with constant $\sum_t t^2\beta_t$, where $\beta_t$ is the density of the squarefree numbers $s_n$ with gap $s_{n+1}-s_n=t$; the paper states (23) for general $\alpha$ with an unnamed constant and says it can prove (23) only for $\alpha$ below a constant between $2$ and $3$. The sketch rests on the sieve bound of Lemma 2 (p. 107), which bounds the number of $s_n<x$ with gap larger than $t$ by a constant times $x/(t^2(\log t)^2)$, so that the tail of the moment sum is small for $\alpha\le2$. Hooley's 1973 paper and Chan's 2023 paper, which extend the range, both cite the result as holding for $0\le\alpha\le2$, and the range stated here is theirs.

Covers. Every exponent $0\le\alpha\le2$. The later ranges are on their own pages: Hooley's $0\le\alpha\le3$ (Hooley 1973), Huxley's $0\le\alpha<11/3$ (Huxley 1997) and Chan's $0\le\alpha<3.75$ (Chan 2023); exponents $\alpha\ge3.75$ are open unconditionally, and Granville derives every $\alpha\ge0$ from the abc conjecture (Granville 1998).

Acceptance. Refereed: Publicationes Mathematicae Debrecen, volume 2 (1951), pp. 103--109. The site labels the problem OPEN (page last edited 19 October 2025), so its curator's remark that Erdős proved the statement for $0\le\alpha\le2$ 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.