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Claim. There is a constant $c>0$ such that for all sufficiently large $x$ the interval $(x,x+cx^{1/5}\log x]$ contains a squarefree number. For the squarefree numbers $s_1<s_2<\cdots$ of Problem 208 this says $s_{n+1}-s_n\ll s_n^{1/5}\log s_n$, so $s_{n+1}-s_n\ll_\epsilon s_n^\epsilon$ for every $\epsilon>1/5$. The paper's single Theorem states the bound; its proof is elementary, counting the integers in a short interval divisible by the square $u^2$ of a large prime through divided differences of $x/u^2$ rather than through exponential sums. The source is M. Filaseta and O. Trifonov, On gaps between squarefree numbers II, J. London Math. Soc. (2) 45 (1992), no. 2, 215--221; its library card is filaseta_1992_gaps_between_squarefree_numbers_ii, with a result page for the Theorem.

Covers. The first question for every $\epsilon>1/5$. Not covered: the first question for $\epsilon\le1/5$, and the second question. A smaller exponent, $1/5-\eta$ for some $\eta>0$, is claimed on Pandey's claim page.

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

Acceptance. Refereed: the paper is the version of record in the Journal of the London Mathematical Society, a refereed journal; the publisher's record dates the issue to April 1992 without a day, so this page is named by the first day of that month. The site's curator credits the bound in the problem's commentary, but the site labels the problem OPEN, so that credit is not acceptance of the problem and no reviewed evidence is listed. The corpus records no check of the proof.