Journal of Integer Sequences, Vol. 23 (2020), Article 20.6.5

The Number of Solutions to ax + by + cz = n and its Relation to Quadratic Residues


Damanvir Singh Binner
Department of Mathematics
Simon Fraser University
Burnaby, BC V5A 1S6
Canada

Abstract:

We develop a formula for the number of non-negative integer solutions (x,y,z) of the equation ax + by + cz = n, where a, b, c, and n are given positive integers. The formula leads us to a surprising connection between the number of non-negative integer solutions of the equation ax + by + cz = n and quadratic residues. As a consequence of our work, we are able to prove the equivalence between two fundamental results by Gauss and Sylvester in the nineteenth century that are generally viewed as independent.


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Received December 21 2017; revised versions received September 25 2019; May 11 2020; May 18 2020. Published in Journal of Integer Sequences, June 11 2020.


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