Problem

Source: German TST 2023, Test 4, Problem 3

Tags: algebra, polynomial, number theory, prime numbers, algebra proposed



Let $f(x)$ be a monic polynomial of degree $2023$ with positive integer coefficients. Show that for any sufficiently large integer $N$ and any prime number $p>2023N$, the product \[f(1)f(2)\dots f(N)\]is at most $\binom{2023}{2}$ times divisible by $p$. Proposed by Ashwin Sah