Problem 208

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claims/: The 3 claim pages of Problem 208, one per claimant's result; the problem's standing derives from them.

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Statement. Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\epsilon>0$ and large $n$,

$$s_{n+1}-s_n \ll_\epsilon s_n^{\epsilon}?$$

Is it true that

$$s_{n+1}-s_n \leq (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}?$$

Status. Open: the site labels the problem OPEN (page last edited 19 October 2025). Neither question is answered. The first is proved for every $\epsilon>1/5$ by Filaseta and Trifonov (1992), claimed for every $\epsilon>1/5-\eta$, with some $\eta>0$, in Pandey's 2024 preprint (a pending partial claim), and proved for every $\epsilon>0$ under the abc conjecture by Granville (1998); the second is open, and Erdős's 1951 lower bound shows that its constant $\pi^2/6$ could not be lowered. The claim pages under claims/ record these results.

Source. erdosproblems.com/208, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #208, https://www.erdosproblems.com/208.

References.

• [Er51] Erdős, P., Some problems and results in elementary number theory. Publ. Math. Debrecen (1951), 103-109.
• [Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.
• [FiTr92] Filaseta, M. and Trifonov, O., On gaps between squarefree numbers II. J. London Math. Soc. (1992), 215-221.
• [Gr98] Granville, Andrew, $ABC$ allows us to count squarefrees. Internat. Math. Res. Notices (1998), 991-1009.
• [Pa24] Pandey, M., Squarefree numbers in short intervals. arXiv:2401.13981 (2024).

Formalization. Statement in formal-conjectures.

Current assessment

The question (site formulation). The two questions above, labeled OPEN, page last edited 19 October 2025. The site's commentary, in this page's words: Erdős [Er51] proved that infinitely many $n$ have $s_{n+1}-s_n>(1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}$, so the bound of the second question, if true, is best possible; in [Er79] Erdős suggests that perhaps $s_{n+1}-s_n\ll\log s_n$ but calls himself very doubtful of it; Filaseta and Trifonov [FiTr92] proved the upper bound $s_n^{1/5+o(1)}$; Pandey [Pa24] lowered the exponent to $1/5-c$ for some $c>0$; and Granville [Gr98] derived $s_{n+1}-s_n\ll_\epsilon s_n^\epsilon$ for every $\epsilon>0$ from the abc conjecture. The site lists Problems 489 and 145 as related and Problem 1101 as a more general form. No forum claim and no AI-assisted result on the problem is recorded.

Claims. The first question asks for the bound $s_{n+1}-s_n\ll_\epsilon s_n^\epsilon$ for every $\epsilon>0$. Unconditionally it is proved for every $\epsilon>1/5$ on Filaseta and Trifonov's claim page (J. London Math. Soc. (2) 45 (1992), 215--221, refereed; an accepted partial claim), and for every $\epsilon>1/5-\eta$, with an unspecified $\eta>0$, on Pandey's claim page (arXiv:2401.13981, a preprint with no publication record, so a pending partial claim). Under the abc conjecture it holds for every $\epsilon>0$ on Granville's claim page (Internat. Math. Res. Notices 1998, 991--1009, refereed; an accepted conditional claim, which settles no standing). The second question has no claim page: Erdős's 1951 theorem is a lower bound for infinitely many $n$, which shows that the constant $\pi^2/6$ could not be lowered but answers neither question, so it has no claim page; the Richert--Rankin bound $n^{2/9+\epsilon}$ that [Er79] reports, and the earlier exponents the [FiTr92] card lists, are superseded by [FiTr92] and are recorded on the cards only. The site labels the problem OPEN, so the curator's commentary is not acceptance of any claim and no page lists reviewed; the corpus records no check of any of the proofs.

Formal statement. The formal-conjectures file at its revision of 2026-09-18 (208.lean) states both questions as separate parts and the $\log s_n$ bound as a variant, all with sorry and the category research open, and names no formal proof.

Search scope. The site's problem page and the formal-conjectures file at the revision above; the library cards of [Er51], [Er79], [FiTr92] and [Pa24]; the publishers' records of [FiTr92] and [Gr98] and the arXiv record of [Pa24]. No wider literature search is recorded, and the openness of the two questions rests on the site's label and these sources.

Remaining gaps. The first question for $\epsilon\le1/5-\eta$ without a hypothesis, and the second question entirely. The library holds no copy of [Gr98], whose result this page records from the site's commentary and the publisher's record.

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.

• pandey_2024_squarefree_numbers_short_intervals
• pandey_2024_squarefree_numbers_short_intervals / theorem_1_1
• erdos_1951_problems_results_elementary_number_theory
• erdos_1951_problems_results_elementary_number_theory / inequality_20
• filaseta_1992_gaps_between_squarefree_numbers_ii
• filaseta_1992_gaps_between_squarefree_numbers_ii / theorem
• erdos_1979_unconventional_problems_number_theory_math_mag