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exist infinitely many non-constant polynomials that are not n-good for any n

Source: Canada Repêchage 2022/4 CMOQR

October 13, 2022
algebrapolynomialInteger Polynomial

Problem Statement

For a non-negative integer nn, call a one-variable polynomial FF with integer coefficients nn-good if: (a) F(0)=1F(0) = 1 (b) For every positive integer cc, F(c)>0F(c) > 0, and (c) There exist exactly nn values of cc such that F(c)F(c) is prime. Show that there exist infinitely many non-constant polynomials that are not nn-good for any nn.
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