Integer solutions to x2 + y2 = z2 −k for a fixed integer value k

Boyer, Wanda; MacGillivray, Gary; Morrison, Laura; Mynhardt, Kieka; Nasserasr, Shahla

doi:10.2140/involve.2017.10.881

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Integer solutions to $x^2 + y^2 = z^2 - k$ for a fixed integer value $k$

Wanda Boyer, Gary MacGillivray, Laura Morrison, C. M. (Kieka) Mynhardt and Shahla Nasserasr

Vol. 10 (2017), No. 5, 881–892

DOI: 10.2140/involve.2017.10.881

Abstract

For a given integer $k$ , general necessary and sufficient conditions for the existence of integer solutions to an equation of the form $x^{2} + y^{2} = z^{2} - k$ are given. It is shown that when there is a solution, there are infinitely many solutions. An elementary method for finding the solutions, when they exist, is described.

Keywords

Diophantine equations, congruences, residue systems, Pythagorean triples

Mathematical Subject Classification 2010

Primary: 11D09

Secondary: 11A07, 11A15

Milestones

Received: 27 July 2016

Accepted: 25 September 2016

Published: 14 May 2017

Communicated by Chi-Kwong Li

Authors

Wanda Boyer
Deparment of Mathematics and Statistics University of Victoria P.O. Box 1700 STN CSC Victoria BC V8W 2Y2 Canada

Gary MacGillivray
Department of Mathematics and Statistics University of Victoria P.O. Box 1700 STN CSC Victoria BC V8W 2Y2 Canada

Laura Morrison
Department of Mathematics and Statistics University of Victoria P.O. Box 1700 STN CSC Victoria BC V8W 2Y2 Canada

C. M. (Kieka) Mynhardt
Department of Mathematics and Statistics University of Victoria P.O. Box 3060 STN CSC Victoria BC V8W 3R4 Canada

Shahla Nasserasr
Department of Mathematics Nova Southeastern University Fort Lauderdale, FL 33324 United States

Vol. 10, No. 5, 2017