#### Vol. 10, No. 5, 2017

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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 ${x}^{2}+{y}^{2}={z}^{2}-k$ 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