Problem 208

Erdős Problem #208: declared status 'open'. Formalized: yes. 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$$s_{n+1}-s_n \leq (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}?$$ Current best: Erd\H{o}s [Er51] showed that there are infinitely many $n$ such that$$s_{n+1}-s_n > (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n},$$so this bound would be the best possible. Filaseta and Trifonov [FiTr92] proved an upper bound of $s_n^{1/5+o(1)}$. Prize: no. OEIS: A005117, A076259. Tags: number theory.