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

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... e 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.