Integer solutions to \(x^2+ y^2= z^2-k\) for a fixed integer value \(k\). (English) Zbl 1364.11077
Summary: 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.
MSC:
| 11D09 | Quadratic and bilinear Diophantine equations |
| 11A07 | Congruences; primitive roots; residue systems |
| 11A15 | Power residues, reciprocity |
References:
| [1] | 10.2307/2689679 · Zbl 0549.10009 · doi:10.2307/2689679 |
| [2] | ; Catalan, Bull. Acad. Roy. Sci. Belgique (3), 9, 531 (1885) |
| [3] | 10.2307/2689346 · Zbl 0626.10014 · doi:10.2307/2689346 |
| [4] | 10.1016/j.laa.2007.10.019 · Zbl 1138.05033 · doi:10.1016/j.laa.2007.10.019 |
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