OPEN
This is open, and cannot be resolved with a finite computation.
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 $\alpha \leq 3$.
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].
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This page was last edited 19 October 2025.
Additional thanks to: Sarosh Adenwalla, Alfaiz, and Wouter van Doorn
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #145, https://www.erdosproblems.com/145, accessed 2026-01-16
