OPEN
Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\alpha \geq 0$,
\[\lim_{x\to \infty}\frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha\]
exists?
Erdős
[Er51] proved this for all $0\leq \alpha \leq 2$, and Hooley
[Ho73] extended this to all $\alpha \leq 3$.
See also [208].
