Problem Statement
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?
Categories:
Number Theory
Progress
Erdős [Er51] proved this for all $0\leq \alpha \leq 2$, and Hooley [Ho73] extended this to all $\alpha \leq 3$.Greaves, Harman, and Huxley showed (in Chapter 11 of [GHH97]) that this is true for $\alpha \leq 11/3$. Chan [Ch23c] has extended this to $\alpha \leq 3.75$.
Granville [Gr98] proved that this follows (for all $\alpha \geq 0$) from the ABC conjecture.
See also [208].
Source: erdosproblems.com/145 | Last verified: January 13, 2026