On the subgroup generated by solutions of Pell's equation

2016/09/02 by Covill, Elena C., Javaheri, Mohammad, Krylov, Nikolai A.
#11A55 #11D09 (Primary) #11R11 #11R20 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1609.00440

Abstract

Equivalence classes of solutions of the Diophantine equation a²+mb²=c² form an infinitely generated abelian group G_m, where m is a fixed square-free positive integer. Solutions of Pell's equation x²-my²=1 generate a subgroup P_m of G_m. We prove that P_m and G_m/P_m have infinite rank for all m>1. We also give several examples of m for which G_m/P_m has nontrivial torsion.

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