Primes dividing values of a given Polynomial

TL;DR

This paper provides a new proof that infinitely many primes divide values of any integer polynomial and extends the result to certain special domains.

Contribution

It introduces a simplified proof of the infinitude of primes dividing polynomial values and generalizes the result to specific algebraic domains.

Findings

Proved the set of primes dividing polynomial values is infinite.

Extended the infinitude result to special algebraic domains.

Provided a new, simpler proof method for classical number theory results.

Abstract

Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. We give an easy and new proof of the fact that the set of primes $p$ such that $p \mid P(n)$, for some $n \in \mathbb{Z}$, is infinite. We also get analog of this result for some special domains.