Diophantine equations

A Diophantine equation asks for integer solutions. For $ax+by=c$, first check that $\gcd (a,b)$ divides $c$; if not, there are no solutions. If it does, find one particular solution $(x_0,y_0)$, then all solutions are $x=x_0+(b/d)t$ and $y=y_0-(a/d)t$ where $d=\gcd (a,b)$ and $t$ is any integer. For $x^2-y^2=n$, factor as $(x-y)(x+y)=n$ and test factor pairs of $n$. For $x^2+y^2=c^2$, test integer squares that sum to $c^2$.