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**GT** on
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For every positive integer n, consider all monic polynomials f(x) with integer coefficients, such that for some real number a

x(f(x+a)−f(x))=nf(x)

Find the largest possible number of such polynomials f(x) for a fixed n<1000.

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial x3+3x−5 is monic but the polynomial −x4+2x3−6 is not.