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Show that there exist polynomials f, g for each n

Source: Iran Pre-Preparation Course Examination 1997, E3, P1

March 28, 2011

algebrapolynomialgreatest common divisoralgebra proposed

Problem Statement

Let

n

be a positive integer. Prove that there exist polynomials

f(x)

and

g(x

) with integer coefficients such that

f(x)\left(x + 1 \right)^{2^n}+ g(x) \left(x^{2^n}+ 1 \right) = 2.