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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
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Abstract |
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For a given integer , general necessary and sufficient conditions for the existence of integer solutions to an equation of the form 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
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Mathematical Subject Classification 2010
Primary: 11D09
Secondary: 11A07, 11A15
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Milestones
Received: 27 July 2016
Accepted: 25 September 2016
Published: 14 May 2017
Communicated by Chi-Kwong Li
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Authors |
| Wanda Boyer | |
| Deparment of Mathematics and
Statistics University of Victoria P.O. Box 1700 STN CSC Victoria BC V8W 2Y2 Canada |
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| Gary MacGillivray | |
| Department of Mathematics and
Statistics University of Victoria P.O. Box 1700 STN CSC Victoria BC V8W 2Y2 Canada |
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| Laura Morrison | |
| Department of Mathematics and
Statistics University of Victoria P.O. Box 1700 STN CSC Victoria BC V8W 2Y2 Canada |
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| C. M. (Kieka) Mynhardt | |
| Department of Mathematics and
Statistics University of Victoria P.O. Box 3060 STN CSC Victoria BC V8W 3R4 Canada |
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| Shahla Nasserasr | |
| Department of Mathematics Nova Southeastern University Fort Lauderdale, FL 33324 United States |
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