# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1901.03653