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Interesting integer polynomial construction

Source: Turkey TST 2022 P6 Day 2

March 13, 2022
algebrapolynomialInteger Polynomialnumber theory

Problem Statement

For a polynomial P(x)P(x) with integer coefficients and a prime pp, if there is no n∈Zn \in \mathbb{Z} such that p∣P(n)p|P(n), we say that polynomial PP excludes pp. Is there a polynomial with integer coefficients such that having degree of 5, excluding exactly one prime and not having a rational root?
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