Cubefree Trinomial Discriminants

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.

arXiv commentThis is an unpublished undergraduate research project and contains errors