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

TL;DR

This paper derives explicit upper bounds for the count of integers up to x where a set of irreducible polynomials simultaneously yield prime values, with applications to Sophie Germain primes relevant to cryptography.

Contribution

It introduces an explicit upper bound method for the number of integers where multiple irreducible polynomials are prime simultaneously, extending previous bounds to arbitrary polynomial sets.

Findings

Explicit upper bounds for simultaneous prime values of polynomial sets

Application to counting Sophie Germain primes up to x

Practical cryptographic implications

Abstract

Using an explicit version of Selberg's upper sieve, we obtain explicit upper bounds for the number of $n\leq x$ 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.