Problem 233

../

---

Statement. Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Prove that

$$\sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2.$$

Status. Open.

Source. erdosproblems.com/233, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #233, https://www.erdosproblems.com/233.

References.

• [Cr36] Cramér, Harald, On the order of magnitude of the difference between consecutive prime numbers. Acta Arithmetica (1936), 23-46.
• [Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp. Section A8 "Gaps between primes. Twin primes.", printed pp. 33--34: Cramér's conditional bound $\sum_{n<x}d_n^2<cx(\ln x)^4$, and "Erdős conjectures that the right-hand side should be $cx(\ln x)^2$, but thinks that there is no hope of a proof". Library home: guy_2004_unsolved_problems_number_theory.
• [Se43] Selberg, Atle, On the normal density of primes in small intervals, and the difference between consecutive primes. Arch. Math. Naturvid. (1943), 87-105.

Formalization. Statement in formal-conjectures.

Current assessment

No current assessment is recorded. The status above is imported from the dated site record. This page records no current literature search or independent assessment of proof coverage.

Known Results

The site's commentary (last edited 18 January 2026) records three bounds, none of which settles the question. Cramér [Cr36] proved, under the Riemann hypothesis, that $\sum_{n\le N}d_n^2\ll N(\log N)^4$. Selberg [Se43] sharpened this slightly, again under the Riemann hypothesis, to $\sum_{n\le N}d_n^2/n\ll(\log N)^4$. In the other direction the prime number theorem gives $\sum_{n\le N}d_n^2\gg N(\log N)^2$, so the conjectured bound would be sharp. The conjectured bound would imply $d_n\ll n^{1/2}\log n$ for every $n$, which is known only under the Riemann hypothesis. Guy [Gu04], Section A8, records the conjecture and Erdős's view that a proof is out of reach. None of these results is a claim on the problem, since none proves or refutes the upper bound.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• guy_2004_unsolved_problems_number_theory
• cramer_1936_order_magnitude_difference_between_consecutive_prime