Involve: A Journal of Mathematics

  • Involve
  • Volume 6, Number 1 (2013), 13-24.

The group of primitive almost pythagorean triples

Nikolai Krylov and Lindsay Kulzer

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We consider the triples of integer numbers that are solutions of the equation x2+qy2=z2, 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{2,3,5,6}. 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.

Article information

Involve, Volume 6, Number 1 (2013), 13-24.

Received: 11 August 2011
Revised: 29 March 2012
Accepted: 29 April 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D09: Quadratic and bilinear equations 20K20: Torsion-free groups, infinite rank

pythagorean triples infinitely generated commutative groups


Krylov, Nikolai; Kulzer, Lindsay. The group of primitive almost pythagorean triples. Involve 6 (2013), no. 1, 13--24. doi:10.2140/involve.2013.6.13.

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