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P_i(a) = N, infinitely many primes

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade,finals p6

November 13, 2022

polynomialalgebra

Problem Statement

Let

n

be a natural number and

P_1, P_2, ... , P_n

are polynomials with integer coefficients, each of degree at least

2

. Let

S

be the set of all natural numbers

N

for which there exists a natural number

a

and an index

1 \le i \le n

such that

P_i(a) = N

. Prove, that there are infinitely many primes that do not belong to

S