Squarefree values of trinomial discriminants

Abstract

The discriminant of a trinomial of the form x n x m 1 has the form n n (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, whenn 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 t into these parametric families. The set of primes whose squares can divide these sporadic values asn varies seems to be independent ofm, 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.