On solving a binary quadratic Diophantine equation

Abstract

We look at methods for solving the Diophantine equation $a{x}^2+bxy+c{y}^2+dx+ey+f=0$ for which $\mathrm{\Delta }=b^2-4ac>0$ and $\mathrm{\Delta }$ is not a square. The methods we use transform this equation to one of the form $A{X}^2+BXY+C{Y}^2=N$. We give upper limits on the number of solutions to the latter equation that need to be reviewed to determine all solutions to the original equation. These upper limits are substantially smaller than those generally given in the literature. We also discuss ways to compactly represent all solutions to the original equation.