Vol. 6, No. 1, 2013

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The group of primitive almost pythagorean triples

Nikolai A. Krylov and Lindsay M. Kulzer

Vol. 6 (2013), No. 1, 13–24
Abstract

We consider the triples of integer numbers that are solutions of the equation ${x}^{2}+q{y}^{2}={z}^{2}$, where $q$ is a fixed, square-free arbitrary positive integer. The set of equivalence classes of these triples forms an abelian group under the operation coming from complex multiplication. We investigate the algebraic structure of this group and describe all generators for each $q\in \left\{2,3,5,6\right\}$. We also show that if the group has a generator with the third coordinate being a power of 2, such generator is unique up to multiplication by $±1$.

Keywords
pythagorean triples, infinitely generated commutative groups
Mathematical Subject Classification 2010
Primary: 11D09, 20K20
Milestones
Revised: 29 March 2012
Accepted: 29 April 2012
Published: 23 June 2013

Communicated by Scott Chapman
Authors
 Nikolai A. Krylov Department of Mathematics Siena College 515 Loudon Road Loudonville, NY 12211 United States Lindsay M. Kulzer Department of Mathematics Siena College 515 Loudon Road Loudonville, NY 12211 United States