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

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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. Minor revision fixing typos, February 1 2021.

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