  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.
