MathDB

Let

f(x)

be a polynomial with positive integer coefficients. For every

n\in\mathbb{N}

, let

a_{1}^{(n)}, a_{2}^{(n)}, \dots , a_{n}^{(n)}

be fixed positive integers that give pairwise different residues modulo

n

and let

g(n) = \sum\limits_{i=1}^{n} f(a_{i}^{(n)}) = f(a_{1}^{(n)}) + f(a_{2}^{(n)}) + \dots + f(a_{n}^{(n)})

Prove that there exists a constant

M

such that for all integers

m>M

we have

\gcd(m, g(m))>2023^{2023}

Problem Statement