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Infinite primes dividing polynomial sequence

Source: STEMS Mathematics 2023 Category B P1

January 8, 2023
algebrapolynomial

Problem Statement

Suppose ff is a nonconstant polynomial with integer coefficients with the following property:
[*]f(0)f(0) and f(1)f(1) are both odd. [*]Define a sequence of integers with ak=f(1)f(2)⋯f(k)+1a_k = f(1)f(2) \cdots f(k)+1
Prove that there are infinitely many prime numbers dividing at least one element of the sequence.
Proposed by Sayandeep Shee
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