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Squarefree values of trinomial discriminants

Part of: Multiplicative number theory Algebraic number theory: global fields

Published online by Cambridge University Press:  01 January 2015

David W. Boyd ,
Greg Martin  and
Mark Thom
David W. Boyd
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2 email boyd@math.ubc.ca
Greg Martin
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2 email gerg@math.ubc.ca
Mark Thom
Affiliation:
Department of Mathematics & Computer Science, University of Lethbridge, C526 University Hall, 4401 University Drive, Lethbridge, AB, Canada T1K 3M4 email markjordanthom@gmail.com

Abstract

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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 $2^{p}$ 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.

MSC classification

Primary: 11N32: Primes represented by polynomials; other multiplicative structure of polynomial values 11N25: Distribution of integers with specified multiplicative constraints 11N05: Distribution of primes
Secondary: 11R29: Class numbers, class groups, discriminants
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 148 - 169
Copyright
© The Author(s) 2015 

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