Problem 145

../

claims/: The 5 claim pages of Problem 145, one per claimant's result; the problem's standing derives from them.

---

Statement. 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?

Status. 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.

Source. erdosproblems.com/145, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #145, https://www.erdosproblems.com/145.

References.

• [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_moments_gaps_between_consecutive_square_free.
• [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: erdos_1951_problems_results_elementary_number_theory.
• [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.

Formalization. Statement in formal-conjectures (pinned at the file's last commit, of 2026-09-18), which tags erdos_145 research open and its variants le_two, le_three and lt_eleven_thirds research solved, each citing the source the site credits (for le_three the site's [Ho73] key, the symposium paper, not the Canadian Journal paper on Hooley's claim page), every proof a sorry and no formal_proof attribute. The ranges are statements, not Lean proofs, and give no formalized evidence.

Current assessment

The site records Problem 145 as OPEN (page last edited 19 October 2025). The question is the existence of the limit, for each $\alpha\ge0$, of the normalized $\alpha$-th moment of the gaps between consecutive squarefree numbers; wherever it is known to exist the limit is $B(\alpha)=\sum_{h\ge1}h^\alpha\alpha(h)$, with $\alpha(h)$ Mirsky's density of the gaps of length $h$, a series that converges for every $\alpha$. The unconditional record is Chan's range $0\le\alpha<3.75$, which subsumes the ranges of Erdős, Hooley and Huxley and the intermediate ranges that Chan's introduction records (Filaseta $29/9$, Filaseta and Trifonov $43/13$, Huxley $59/16$; not credited by the site and not paged, being subsumed). The endpoint $\alpha=3.75$ and every larger exponent are open unconditionally; Granville's paper derives every $\alpha\ge0$ from the abc conjecture. The claimed statements are checked against the papers' abstracts, Chan's introduction and theorem, and the Erdős card only; no proof is rechecked, and no literature search goes beyond the sources the site names. Chan's paper says that further progress depends on improving Huxley's treatment of the gaps built from many squares of mid-sized primes.

Known Results

• Erdős 1951: the moment asymptotic for $0\le\alpha\le2$, accepted on refereed publication.
• Hooley 1973: the range $0\le\alpha\le3$, accepted on refereed publication.
• Huxley 1997: the range $0\le\alpha<11/3$, pending, from a proceedings volume.
• Chan 2023: the range $0\le\alpha<3.75$, accepted on refereed publication; the paper has a library card.
• Granville 1998: every $\alpha\ge0$ under the abc conjecture, a conditional claim.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• chan_2023_moments_gaps_between_consecutive_square_free
• erdos_1951_problems_results_elementary_number_theory
• erdos_1951_problems_results_elementary_number_theory / equation_23
• erdos_1951_problems_results_elementary_number_theory / lemma_2