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All Random Solved Random Open
OPEN
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 \[s_{n+1}-s_n \leq (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}?\]
Erdős [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 $s_n^{1/5}$. Pandey [Pa24] has improved this exponent to $1/5-c$ for some constant $c>0$.

See also [489].

Additional thanks to: Zachary Chase