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Claim. Assume the abc conjecture. Then for every $\epsilon>0$ the squarefree numbers $s_1<s_2<\cdots$ of Problem 208 satisfy $s_{n+1}-s_n\ll_\epsilon s_n^\epsilon$, the bound the first question asks for. The site's commentary credits the paper with this deduction, and the paper's title states its theme, that the abc conjecture lets one count the squarefree values of polynomials and the squarefree numbers in short intervals. The source is A. Granville, ABC allows us to count squarefrees, Internat. Math. Res. Notices 1998, no. 19, 991--1009; the library holds no copy, and this page records the result from the site's commentary and the publisher's record.

Hypothesis. The abc conjecture: for every $\epsilon>0$ there is a constant $K_\epsilon$ such that every triple of coprime positive integers with $a+b=c$ satisfies $c\le K_\epsilon\,\mathrm{rad}(abc)^{1+\epsilon}$, where $\mathrm{rad}(m)$ is the product of the distinct primes dividing $m$. It is unproved, and the claim gives no unconditional answer.

Scope. The claim is conditional and settles no standing of the problem by itself. It concerns the first question only; the second question, the bound $(1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}$, is not addressed. Unconditionally the first question is known for every $\epsilon>1/5$ (Filaseta and Trifonov) and claimed for every $\epsilon>1/5-\eta$ (Pandey).

Depends on. Nothing in this wiki; the hypothesis is stated above.

Acceptance. Refereed: International Mathematics Research Notices is a refereed journal, and the publisher's record dates the article to 1998 without a month or day, so this page is named by the first day of that year. The site labels the problem OPEN, so the curator's credit is not acceptance and no reviewed evidence is listed. The corpus records no check of the proof.