Problem

Source: RMO - problem 4

Tags: algebra, polynomial, quadratics, modular arithmetic, number theory unsolved, number theory



A polynomial is called Fermat polynomial if it can be written as the sum of squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0)=1000$. Prove that $f(x)+2x$ is not a fermat polynomial