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arXiv · 1910.03079

Number of Non-negative Integer Solutions of the Equation ax+by+cz=n and its Relation with Quadratic Residues

Damanvir Singh Binner
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Abstract

In 2000, A. Tripathi used generating functions to obtain a formula for the number of non-negative solutions (x,y) of the equation ax + by = n where a, b and n are given positive integers. We generalize this procedure for the number of solutions of the equation ax + by + cz = n. The formula leads us to a surprising connection between the number of 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 which are generally viewed to be independent.

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Damanvir Singh Binner. 2019-09-23. Number of Non-negative Integer Solutions of the Equation ax+by+cz=n and its Relation with Quadratic Residues. https://arxiv.org/abs/1910.03079

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