Cubefree Trinomial Discriminants

TL;DR

This paper studies the discriminants of specific trinomial polynomials, characterizing when they have prime power divisors, and proves the existence of infinitely many cases with no prime cube divisors.

Contribution

It provides a classification of discriminants for polynomials of the form ±x^n ± x^m ± 1 and proves the existence of infinitely many pairs with discriminants free of prime cube divisors.

Findings

Discriminants have specific forms when n,m are coprime.

There are infinitely many pairs (n,m) with discriminants having no prime cube divisors.

Symmetries in the discriminant values are identified and explained.

Abstract

The discriminant of a polynomial of the form $\pm x^n \pm x^m \pm 1$ has the form $n^n \pm m^m(n-m)^{n-m}$ when $n,m$ are relatively prime. We investigate when these discriminants have prime power divisors. We explain several symmetries that appear in the classification of these values of $n,m$. We prove that there are infinitely many pairs of integers $n,m$ for which this discriminant has no prime cube divisors. This result is extended to show that for infinitely many fixed $m$, there are infinitely many $n$ for which the discriminant has no prime cube divisor.