On the $P_1$ property of sequences of positive integers

TL;DR

This paper presents two analytic criteria to prove that sequences generated by non-constant integer polynomials have infinitely many prime divisors, offering alternative proofs to classical results.

Contribution

It introduces new analytic methods for establishing the $P_1$ property of polynomial sequences, expanding the toolkit beyond traditional algebraic proofs.

Findings

Two analytic criteria established for the $P_1$ property

Alternative proofs for polynomial sequences with infinitely many prime divisors

Enhanced understanding of prime divisibility in polynomial-generated sequences

Abstract

It is well-known that for any non-constant polynomial $P$ with integer coefficients the sequence $(P(n))_{ n\in \mathbb N}$ has the property that there are infinitely many prime numbers dividing at least one term of this sequence. Certainly, there is a proof based on the Chinese Remainder Theorem. In this paper we give proofs of two analytic criteria revealing this property of sequences.