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
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
Full version: pdf,
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. Minor revision fixing typos, February 1 2021.
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