Problem 145

Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\alpha \geq 0$ ,

\lim_{x\to \infty}\frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha

exists?

Source: erdosproblems.com/145

Number theory

Wiki page

Status

Open

No claim settles this problem.

Open. The site labels the problem OPEN (page last edited 19 October 2025) and its commentary credits a chain of partial results, each recorded as a claim page: Erdős's range $0\le\alpha\le2$ (Erdős 1951), Hooley's $0\le\alpha\le3$ (Hooley 1973), Huxley's $0\le\alpha<11/3$ (Huxley 1997, pending, since the proceedings volume has no refereeing on record) and Chan's $0\le\alpha<3.75$ (Chan 2023), and Granville's derivation of every $\alpha\ge0$ from the abc conjecture (Granville 1998, conditional). Every exponent $\alpha\ge3.75$ is open unconditionally, so no full claim exists and the frontmatter standing stays open.

Ch23c: Chan, Tsz Ho, On moments of gaps between consecutive square-free numbers. Mosc. J. Comb. Number Theory 12 (2023), no. 4, 287-295, doi:10.2140/moscow.2023.12.287; arXiv:2310.08448. Library home: Chan 2023.
Er51: Erdős, P., Some problems and results in elementary number theory. Publ. Math. Debrecen 2 (1951), 103-109, doi:10.5486/pmd.1951.2.2.04. Library home: Erdős 1951.
GHH97: Greaves, G. R. H. and Harman, G. and Huxley, M. N. (eds.), Sieve Methods, Exponential Sums, and their Applications in Number Theory (Cardiff, 1995). London Math. Soc. Lecture Note Ser. 237, Cambridge Univ. Press (1997), doi:10.1017/CBO9780511526091. The result the site credits under this key is Chapter 11, Huxley, M. N., Moments of differences between square-free numbers, pp. 187-204, doi:10.1017/CBO9780511526091.014, a chapter by Huxley alone.
Gr98: Granville, Andrew, $ABC$ allows us to count squarefrees. Internat. Math. Res. Notices 1998, no. 19, 991-1009, doi:10.1155/S1073792898000592.
Ho73: Hooley, Christopher, On the intervals between consecutive terms of sequences. Proc. Sympos. Pure Math. 24 (1973), 129-140, doi:10.1090/pspum/024/0384742. The site's commentary credits the range $\alpha\le3$ to Hooley under this key, which the site's reference record resolves to this symposium paper; the paper that proves the range is Hooley, C., On the distribution of square-free numbers, Canad. J. Math. 25 (1973), no. 6, 1216-1223, doi:10.4153/CJM-1973-129-0, whose abstract states the result and which Chan's reference list cites for it. The claim page cites the Canadian Journal paper.

Problem 145

Status

Proof claims

Discussion