The Diophantine equation x^2 + y^2 = p can be solved for p a prime iff p congruent 1 (mod 4) or p = 2. The representation is unique except for changes of sign or rearrangements of x and y. This theorem is intimately connected with the quadratic reciprocity theorem, and generalizes to the biquadratic reciprocity theorem.

composition theorem | Diophantine equation--4th powers | Fermat's theorem | form genus | fundamental theorem of genera | quadratic reciprocity theorem