2019/09/23 by Damanvir Singh Binner, Binner, Damanvir Singh · 1 citation
Mathematics · #History and Theory of Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research
paper · pdf · doi:10.48550/arxiv.1910.03079
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.