# How many polynomials?

For every positive integer $n,$ consider all monic polynomials $f(x)$ with integer coefficients, such that for some real number $a$ $x\left[ f(x+a)-f(x) \right]=nf(x)$ 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 $x^3 + 3x - 5$ is monic but the polynomial $-x^4 + 2x^3 - 6$ is not.

×