Vol. 10, No. 5, 2017

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ISSN: 1944-4184 (e-only)
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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
Abstract

For a given integer k, general necessary and sufficient conditions for the existence of integer solutions to an equation of the form x2 + y2 = z2 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