  The discriminant of a trinomial of the form $x^n \pm x^m \pm 1$ has the form
$\pm n^n \pm (n-m)^{n-m} m^m$ if $n$ and $m$ are relatively prime. We
investigate when these discriminants have nontrivial square factors. We explain
various unlikely-seeming parametric families of square factors of these
discriminant values: for example, when $n$ is congruent to 2 (mod 6) we have
that $((n^2-n+1)/3)^2$ always divides $n^n - (n-1)^{n-1}$. In addition, we
discover many other square factors of these discriminants that do not fit into
these parametric families. The set of primes whose squares can divide these
sporadic values as $n$ varies seems to be independent of $m$, and this set can
be seen as a generalization of the Wieferich primes, those primes $p$ such that
$2^{p-1}$ is congruent to 1 (mod $p^2$). We provide heuristics for the density
of squarefree values of these discriminants and the density of these "sporadic"
primes.
