Squarefree values of trinomial discriminants

Abstract

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.

22 pages, 1 table. Minor revisions from version 1, including three new references

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