Experimental Mathematics

Frequencies of Successive Pairs of Prime Residues

Avner Ash, Laura Beltis, Robert Gross, and Warren Sinnott

Full-text: Open access


We consider statistical properties of the sequence of ordered pairs obtained by taking the sequence of prime numbers and reducing modulo $m$. Using an inclusion/exclusion argument and a cutoff of an infinite product suggested by Pólya, we obtain a heuristic formula for the "probability" that a pair of consecutive prime numbers of size approximately $x$ will be congruent to $(a, a + d)$ modulo $m$. We demonstrate some symmetries of our formula. We test our formula and some of its consequences against data for $x$ in various ranges.

Article information

Experiment. Math., Volume 20, Issue 4 (2011), 400-411.

First available in Project Euclid: 8 December 2011

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11N05: Distribution of primes 11K45: Pseudo-random numbers; Monte Carlo methods 11N69: Distribution of integers in special residue classes

Prime pairs Bateman–Horn Hardy–Littlewood Pólya


Ash, Avner; Beltis, Laura; Gross, Robert; Sinnott, Warren. Frequencies of Successive Pairs of Prime Residues. Experiment. Math. 20 (2011), no. 4, 400--411. https://projecteuclid.org/euclid.em/1323367154

Export citation