Journal of Integer Sequences, Vol. 23 (2020), Article 20.9.3

On the First Occurrences of Gaps Between Primes in a Residue Class

Alexei Kourbatov
Redmond, WA 98052

Marek Wolf
Faculty of Mathematics and Natural Sciences
Cardinal Stefan Wyszyński University
Warsaw, PL-01-938


We study the first occurrences of gaps between primes in the arithmetic progression (P): r, r + q, r + 2q, r + 3q, . . . , where q and r are coprime integers, q > r ≥ 1. The growth trend and distribution of the first-occurrence gap sizes are similar to those of maximal gaps between primes in (P). The histograms of first-occurrence gap sizes, after appropriate rescaling, are well approximated by the Gumbel extreme value distribution. Computations suggest that first-occurrence gaps are much more numerous than maximal gaps: there are O(log2 x) first-occurrence gaps between primes in (P) below x, while the number of maximal gaps is only O(log x). We explore the connection between the asymptotic density of gaps of a given size and the corresponding generalization of Brun's constant. For the first occurrence of gap d in (P), we expect the end-of-gap prime p ≍ √d exp√d/φ(q) infinitely often. Finally, we study the gap size as a function of its index in the sequence of first-occurrence gaps.

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(Concerned with sequences A000010 A001620 A005597 A014320 A065421 A167864 A194098 A268925 A268928 A330853 A330854 A330855 A334543 A334544 A334545 A335366 A335367.)

Received June 11 2020; revised versions received September 15 2020, September 19 2020. Published in Journal of Integer Sequences, October 15 2020.

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