Problem Statement
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 $q_1,q_2\leq p$?
Categories:
Number Theory Primes
Progress
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].
Source: erdosproblems.com/17 | Last verified: January 13, 2026