OPEN
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 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 $c<1/8$.
