Strings of congruent primes in short intervals

TL;DR

This paper proves the existence of infinitely many pairs of consecutive primes in short intervals that are congruent modulo q, extending understanding of prime distribution in arithmetic progressions.

Contribution

It introduces a novel proof combining ideas from Shiu and Goldston-Pintz-Yildirim to establish the infinitude of such prime pairs in short intervals.

Findings

Infinitely many prime pairs with the same residue modulo q in short intervals.

Prime gaps less than epsilon times log p_r occur infinitely often.

Extension of prime distribution results in arithmetic progressions.

Abstract

Fix \epsilon > 0, and let p_1 = 2, p_2 = 3,... be the sequence of all primes. We prove that if (q,a) = 1 then there are infinitely many pairs p_r, p_{r+1} such that p_r \equiv p_{r+1} \equiv a \mod q and p_{r+1} - p_r < \epsilon\log p_r. The proof combines the ideas of Shiu and Goldston-Pintz-Yildirim.