  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.
