Finding

Paper

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.

Authors

D. Boyd, G. Martin, Mark Thom

Journal

Lms Journal of Computation and Mathematics

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