Explicit upper bounds for the number of primes simultaneously representable by any set of irreducible polynomials

2022/11/20 by Matteo Bordignon, Bordignon, Matteo, Ethan Simpson Lee +1
Computer Science · Mathematics · #11N32 #11N35 #11N36 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2211.11012

openalex publication_date 2022/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using an explicit version of Selberg's upper sieve, we obtain explicit upper bounds for the number of n≤ x such that a non-empty set of irreducible polynomials F_i(n) with integer coefficients are simultaneously prime; this set can contain as many polynomials as desired. To demonstrate, we present computations for some irreducible polynomials and obtain an explicit upper bound for the number of Sophie Germain primes up to x, which have practical applications in cryptography.

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