Genus Theorem

A Diophantine Equation

$$x^2+y^2=p$$

can be solved for $p$ a Prime Iff $p\equiv 1\ \left({{\rm mod\ } {4}}\right)$ or $p=2$. The representation is unique except for changes of sign or rearrangements of $x$ and $y$.

See also Composition Theorem, Fermat's Theorem