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Statement

Theorem 2 (p. 45). "The sum of the reciprocals of the cluster primes is finite."

Cluster primes are as in the definition of p. 43. The statement holds trivially if there are only finitely many of them; the paper does not decide whether there are.

Source. R. Blecksmith, P. Erdős and J. L. Selfridge, Cluster primes, Amer. Math. Monthly 106 (1999), no. 1, 43--48; Theorem 2 and its proof on p. 45, read on the page image of the copy identified on the source card.

Read depth. Claims checked: the statement was read on the page image and the short proof was read through. Nothing here is independently reviewed.

Proof pointer

Page 45. Assume there are infinitely many cluster primes and let $q_n$ be the $n$-th. By Theorem 1 with $s=2$, $n=\pi_c(q_n)<q_n/(\log q_n)^2$ for large $n$, and since $(\log q_n)^2>(\log n)^2$ this gives $q_n>n(\log n)^2$. The series $\sum n^{-1}(\log n)^{-2}$ converges, so $\sum1/q_n$ converges by comparison. The paper notes (p. 48) that this is essentially Brun's 1921 argument for the twin primes, run from the bound $\pi_2(x)\ll x/(\log x)^2$ that Lemma 2 gives with $s=1$, $d_1=2$.

Dependencies

Theorem 1 with $s=2$.

Bears on

• Problem 17: the problem's primes are sparse enough that their reciprocals have a finite sum. This does not decide whether there are infinitely many of them.