Fermat's Theorem

A Prime $p$ can be represented in an essentially unique manner in the form $x^2+y^2$ for integral $x$ and $y$ Iff $p\equiv 1\ \left({{\rm mod\ } {4}}\right)$ or $p=2$. It can be restated by letting

$$Q(x,y)\equiv x^2+y^2,$$

then all Relatively Prime solutions $(x,y)$ to the problem of representing $Q(x,y)=m$ for $m$ any Integer are achieved by means of successive applications of the Genus Theorem and Composition Theorem. There is an analog of this theorem for Eisenstein Integers.

See also Eisenstein Integer, Square Number

References

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 142-143, 1993.