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Existence of a polynomial

Source: Finnish Mathematics Competition 2007, Final Round, Problem 5

November 14, 2011
algebrapolynomialmodular arithmeticalgebra unsolved

Problem Statement

Show that there exists a polynomial P(x)P(x) with integer coefficients, such that the equation P(x)=0P(x) = 0 has no integer solutions, but for each positive integer nn there is an x∈Zx \in \Bbb{Z} such that n∣P(x).n \mid P(x).
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