Blecksmith 1999 cluster primes

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conjecture_p45: The paper's unproved conjecture that for some constant alpha the number of cluster primes up to x is at most a constant times x divided by e^{alpha (log log x)^2}, stronger than Theorem 1.

definition_p43: Blecksmith, Erdős and Selfridge's definition of a cluster prime, a prime p greater than 2 such that every even positive integer below p minus 2 is a difference of two primes not exceeding p, and their question whether there are infinitely many.

theorem_1: For every positive integer s there is x_0(s) such that the number of cluster primes not exceeding x is less than x/(log x)^s for all x at least x_0(s), proved with Brun's sieve.

theorem_2: The sum of the reciprocals of the cluster primes converges, deduced from Theorem 1 with s equal to 2.

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R. Blecksmith, P. Erdős and J. L. Selfridge, Cluster primes, Amer. Math. Monthly 106 (1999), no. 1, 43--48; JSTOR stable URL http://www.jstor.org/stable/2589585; DOI 10.1080/00029890.1999.12005005 (its Crossref record gives the same volume, issue and pages).

The copy read for this card is the JSTOR PDF: a cover sheet with a text layer (physical p. 1, giving the bibliographic data and stable URL) followed by image-only scans of the six printed pages 43--48 (physical p. $n$ is printed p. $41+n$ for $n\ge2$). The statements below were read on the page images. Provenance: a September 2026 download whose URL was not recorded, though the cover sheet names the stable URL above. 153,658 bytes. Its JSTOR cover sheet prints "you may use content in the JSTOR archive only for your personal, non-commercial use" and "Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page" and names the publisher as the Mathematical Association of America, while the image-only article page printed p. 43 shows no copyright line, every other right reserved.

Read status: claims checked. The definition, the question of p. 43, Theorem 1, Theorem 2 and the Conjecture of p. 45 were read clause by clause on the page images; the proofs were read but not checked.

Contents

• Definition (p. 43): a prime $p>2$ is a cluster prime when each even number $2r$ with $0<2r<p-2$ equals $q-q'$ for some primes $q,q'\le p$. The first 23 odd primes $3,\dots,89$ are cluster primes and 97 is the smallest non-cluster prime ($88$ is not a difference of primes below 98). Page 43 asks: "Are there infinitely many cluster primes?" A positive answer would give $p_{n+1}-p_n\le6$ for infinitely many $n$. This is the question of #17.
• Theorem 1 (p. 44; proof pp. 44--45): for every positive integer $s$ there is $x_0=x_0(s)$ such that $\pi_c(x)<x/(\log x)^s$ for $x\ge x_0$, where $\pi_c(x)$ counts cluster primes up to $x$. The proof uses Lemma 1 ($\pi(x)<(2x-6)/\log x$ for $x\ge6$, from Rosser and Schoenfeld) to show that a cluster prime $p$ has at least $\frac14\log t$ primes in $[p-t,p)$, and Lemma 2 (Brun's sieve, from Halberstam and Richert) to bound the number of $p\le x$ with $s$ prescribed prime differences by $Mx/(\log x)^{s+1}$. The acknowledgments (p. 48) record that the proof is Erdős's handwritten one, with Halberstam helping to elucidate the phrase "by Brun's sieve". Page 48 remarks that $x_0(s)$ is astronomically large.
• Theorem 2 (p. 45): "The sum of the reciprocals of the cluster primes is finite." It is deduced from Theorem 1 with $s=2$.
• Conjecture (p. 45): for some constant $\alpha$, $\pi_c(x)\ll x/e^{\alpha(\log\log x)^2}$ (4); it would follow from Lemma 2 if its implied constant did not grow too fast with $s$.
• Section 3 (pp. 45--46): an algorithm that finds the next cluster prime without backtracking. Section 4 (pp. 46--48): counts up to $10^k$, for $2\le k\le13$, of the cluster primes, the non-cluster primes and the twin-prime pairs ($\pi_c(10^{13})=1{,}061{,}375{,}739$, non-cluster primes outnumbering them about 325 to 1), the value of $\alpha$ that makes (4) an equality at each $10^k$ (0.6301 at $10^2$, 0.7921 at $10^{13}$), the longest run of 10,543 consecutive non-cluster primes found below $10^{13}$, and a comparison with the twin primes, whose Brun bound $x/(\log x)^2$ is weaker than Theorem 1, though the paper cautions (p. 48) that $x_0(s)$ is far beyond the computed range and that both sets could still be finite.

Compiled scope

The statements above were read on the page images (pp. 43--45 in full, pp. 46--48 for the data cited). The proofs of Theorems 1 and 2 were read but not checked, and Lemma 2's sieve input was not consulted. Nothing here is independently reviewed.

Bears on. #17, as the paper that defines the problem's primes and poses its question (Definition, p. 43), proves that fewer than $x/(\log x)^s$ of them are at most $x$ for each fixed $s$ and large $x$ (Theorem 1) and that their reciprocals have a finite sum (Theorem 2), and conjectures a sharper upper bound (Conjecture); it does not decide whether there are infinitely many cluster primes.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.