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1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x), integer pol.

Source: Balkan MO Shortlist 2013 A4 BMO

March 9, 2020
Integer Polynomialpolynomialalgebra

Problem Statement

Find all positive integers nn such that there exist non-constant polynomials with integer coefficients f1(x),...,fn(x)f_1(x),...,f_n(x) (not necessarily distinct) and g(x)g(x) such that 1+∏k=1n(fk2(x)−1)=(x2+2013)2g2(x)1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x)
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