Are there infinitely many cluster primes

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

An odd prime $p$ is a cluster prime if, for every even positive integer $d$ with $d\le p-3$, there exist primes $q,r\le p$ with $q-r=d$. The prime $2$ is excluded by convention; for $p=3$ the condition is empty.

Statement recorded in the supplied source.

Are there infinitely many cluster primes? Equivalently, is the set of odd primes satisfying this universal difference condition unbounded? [S2]

Short English statement

Do infinitely many odd primes have the property that every possible positive even difference up to three less than the prime is realized by two primes no larger than it?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400145 (AMR-093-0145). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, Cluster prime, revision 1377809312 (2026) Lead and infinitude question. https://en.wikipedia.org/w/index.php?title=Cluster+prime&oldid=1377809312.