Squarefree values of trinomial discriminants

Abstract

AbstractThe 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 $2^{p}$ is congruent to 2 (mod $p^{2}$). We provide heuristics for the density of these sporadic primes and the density of squarefree values of these trinomial discriminants.