OPEN
This is open, and cannot be resolved with a finite computation.
Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Prove that\[\sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2.\]
Cramer
[Cr36] proved an upper bound of $O(N(\log N)^4)$ conditional on the Riemann hypothesis. Selberg
[Se43] improved this slightly (still assuming the Riemann hypothesis) to\[\sum_{1\leq n\leq N}\frac{d_n^2}{n}\ll (\log N)^4.\]The prime number theorem immediately implies a lower bound of\[\sum_{1\leq n\leq N}d_n^2\gg N(\log N)^2.\]The values of the sum are listed at
A074741 on the OEIS.
This is discussed in problem A8 of Guy's collection
[Gu04].
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This page was last edited 28 October 2025.
Additional thanks to: Dogmachine and Terence Tao
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