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Statement

Definition (p. 43). "A prime $p>2$ is called a cluster prime if every even positive integer less than $p-2$ can be written as a difference of two primes $q-q'$, where $q$ and $q'$ are both less than or equal to $p$."

Equivalently, every even $n$ with $2\le n\le p-3$ is $q-q'$ with $q,q'$ primes at most $p$. The paper writes $\pi_c(x)$ for the number of cluster primes not exceeding $x$ (p. 43), and does not count $2$ as either a cluster or a non-cluster prime (p. 47).

Facts recorded with the definition (p. 43).

• The first $23$ odd primes $3,5,7,\dots,89$ are cluster primes, and $97$ is the smallest non-cluster prime: the previous prime is $89$, and $88=97-9$ is not a difference of two primes below $98$.
• If $p$ is a cluster prime, then the even numbers $p-9$, $p-15$, $p-21$, $p-25$, and so on must all be differences of primes below $p$, so $p$ has enough primes in a short interval to its left.
• The first twin pair $\{p-2,p\}$ with $p$ not a cluster prime is $\{227,229\}$: $202=229-27$ is not a difference of primes below $230$.

The question (p. 43), as posed: "Are there infinitely many cluster primes?" The paper notes that an affirmative answer would imply $p_{n+1}-p_n\le6$ for infinitely many primes $p_n$, which it calls a well-known hopeless problem.

Source. R. Blecksmith, P. Erdős and J. L. Selfridge, Cluster primes, Amer. Math. Monthly 106 (1999), no. 1, 43--48; the definition, the small cases and the question on p. 43, the convention on $2$ on p. 47, read on the page images of the copy identified on the source card.

Read depth. Claims checked: the definition and the question were read clause by clause on the page image. The small-case facts were read as printed and not recomputed. Nothing here is independently reviewed.

Proof pointer

None needed for the definition. The small cases are checks on the primes up to $230$ stated on p. 43.

Dependencies

None.

Bears on

• Problem 17: the problem's primes are the cluster primes defined here, and its question is the question of p. 43. The problem's range $n\le p-3$ for even $n$ is the definition's range of even positive integers less than $p-2$. The paper poses the question and does not answer it.