On the $P_1$ property of sequences of positive integers

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.