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Self dividing polynomial

Source: 2023 Kürschák Mathematics Competition/1

October 7, 2023
number theorypolynomialalgebra

Problem Statement

Let f(x)f(x) be a non-constant polynomial with non-negative integer coefficients. Prove that there are infinitely many positive integers nn, for which f(n)f(n) is not divisible by any of f(2)f(2), f(3)f(3), ..., f(n1)f(n-1).