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Statement

Conjecture (p. 45), as posed: "For some constant $\alpha$, we have

$$\pi_c(x)\ll\frac{x}{e^{\alpha(\log\log x)^2}}.$$

"

This is display (4). Here $\pi_c(x)$ counts the cluster primes not exceeding $x$ (see the definition of p. 43), and $f(x)\ll g(x)$ means that for some constant $M$ and some $x_0$, $|f(x)|\le Mg(x)$ for all $x\ge x_0$ (p. 44). The paper introduces it as a possibly stronger result than Theorem 1 and says it would follow from Lemma 2 of p. 44 if the constant implied there did not grow too fast with $s$. It is not proved in the paper.

Data (p. 47). The table of p. 47 gives, for $x=10^k$ with $2\le k\le13$, the value $\alpha=\log(x/\pi_c(x))/(\log\log x)^2$ at which (4) becomes an equality; it is $0.6301$ at $10^2$ and $0.7921$ at $10^{13}$, where $\pi_c(10^{13})=1{,}061{,}375{,}739$.

Source. R. Blecksmith, P. Erdős and J. L. Selfridge, Cluster primes, Amer. Math. Monthly 106 (1999), no. 1, 43--48; the conjecture on p. 45, the $\ll$ notation on p. 44, the table on p. 47, read on the page images of the copy identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page image, and the two cited table entries were read as printed. Nothing here is independently reviewed.

Proof pointer

None: the paper states it as a conjecture.

Dependencies

None proved. The paper's suggested route is Lemma 2 of p. 44 (Brun's sieve) with control of its implied constant in $s$.

Bears on

• Problem 17: a conjectured sharper upper bound for the number of the problem's primes up to $x$. Like Theorem 1, it would not decide whether there are infinitely many.