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Polynomial having infinitely many prime divisors

Source: Iran 3rd round 2011-Number Theory exam-P1

September 19, 2012
algebrapolynomialmodular arithmeticinequalitiesnumber theoryprime numbersnumber theory proposed

Problem Statement

P(x)P(x) is a nonzero polynomial with integer coefficients. Prove that there exists infinitely many prime numbers qq such that for some natural number nn, q∣2n+P(n)q|2^n+P(n).
Proposed by Mohammad Gharakhani
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