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OPEN This is open, and cannot be resolved with a finite computation.
Are there infinitely many primes $p$ such that every even number $n\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $q_1,q_2\leq p$?
The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
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The first prime without this property is $97$. The sequence of such primes is A038133 in the OEIS. These are called cluster primes.

Blecksmith, Erdős, and Selfridge [BES99] proved that the number of such primes is\[\ll_A \frac{x}{(\log x)^A}\]for every $A>0$, and Elsholtz [El03] improved this to\[\ll x\exp(-c(\log\log x)^2)\]for every $c<1/8$.

This is discussed in problem C1 of Guy's collection [Gu04].

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This page was last edited 28 December 2025. View history

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Formalised statement? Yes
Related OEIS sequences: A038133
1 comment on this problem
Likes this problem Dogmachine, holyterror
Interested in collaborating None
Currently working on this problem None
This problem looks difficult TerenceTao
This problem looks tractable None
The results on this problem could be formalisable None
I am working on formalising the results on this problem None

Additional thanks to: Ralf Stephan and Terence Tao

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 #17, https://www.erdosproblems.com/17, accessed 2026-05-21
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  • Obviously related to the question of which numbers can be written as difference of primes. Paulo Ribenboim once remarked that it has never been proved that every even number can be written as difference of primes.

    Dogmachine — 18:36 on 09 Aug 2025

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