Erdős Problem #145

Canonical assertion

declared status 'open'. Formalized: yes. 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? Current best: Erd\H{o}s [Er51] proved this for all $0\leq \alpha \leq 2$, and Hooley [Ho73] extended this to all $\alpha \leq 3$. Prize: no. OEIS: A005117. Tags: number theory.