2nd January 2026

AMM Problem 12551 Solution

Number Theory: On the orders of $mod p$ roots of integer polynomials

In this article, I provide the solution to AMM Problem 12551. This article is being published a day after the AMM deadline!
This is part of my December 31st 2025 deadline, of which I solved 5/7 problems that I'm sharing here! This is the 5th and final such article.

This problem is a number theory based problem about the orders of roots of polynomials $\mod p$ .
We recall the definition for $ord_p(r) =$ the minimum power $k \ge 0$ of $r$ such that $r^k \equiv 1 \mod p$ .
For any polynomial $f \in \mathbb{Z}[x]$ with $f(0) \ne 0$ , we define

$S_f = \left\{ ord_p(r): f(r) \equiv 0 \mod p, p \text{ is a prime}, r \in (\mathbb{Z}/p\mathbb{Z})^{\times} \right\}$

The problem asks us to prove that

S_f

is finite iff

f | (x^d-1)^c

for some positive integers

c,d

.
The problem is deeply rooted in both number theory and algebra, requiring us to recall polynomial division, as well as properties of congruence.
Overall, quite interesting!

I hope you enjoyed reading the article

Any feedback is greatly appreciated!