On the Infinitude of Some Special Kinds of Primes

TL;DR

This paper explores conditions under which multivariable number-theoretic functions simultaneously produce infinitely many primes, extending classical results and proposing new conjectural frameworks.

Contribution

It introduces new necessary conditions for prime representation by polynomials, generalizes classical functions, and proposes a unified model for prime-producing functions.

Findings

Identified new necessary conditions for prime-generating polynomials.

Proposed an analogy of the Chinese Remainder Theorem.

Generalized Euler's totient and prime-counting functions.

Abstract

The aim of this paper is to try to establish a generic model for the problem that several multivariable number-theoretic functions represent simultaneously primes for infinitely many integral points. More concretely, we introduced briefly the research background-the history and current situation-from Euclid's second theorem to Green-Tao theorem. We analyzed some equivalent necessary conditions that irreducible univariable polynomials with integral coefficients represent infinitely many primes, found new necessary conditions which perhaps imply that there are only finitely many Fermat primes, obtained an analogy of the Chinese Remainder Theorem, generalized Euler's function, the prime-counting function and Schinzel-Sierpinski's Conjecture and so on. Nevertheless, this is only a beginning and it miles to go. We hope that number theorists consider further it.