Strings of congruent primes in short intervals II

2011/10/30 by Tristan Freiberg, Freiberg, Tristan
Computer Science · Mathematics · #11N05 (Primary) #11N13 (Secondary) #Cellular Automata and Applications #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1110.6624

openalex publication_date 2011/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p₁ = 2, p₂ = 3,... be the sequence of all primes. Let ε be an arbitrarily small but fixed positive number, and fix a coprime pair of integers q ≥ 3 and a. We will establish a lower bound for the number of primes p_r, up to X, such that both p_r+1 - p_r < εlog p_r and p_r ≡ p_r+1 ≡ a \bmod q simultaneously hold. As a lower bound for the number of primes satisfying the latter condition, the bound we obtain improves upon a bound obtained by D. Shiu.

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