Bruck-Ryser theorem

Statement
If there exists a fact about::projective plane of finite order $$q$$ (or equivalently, there exists an fact about::affine plane of order $$q$$) and if $$q$$ is congruent to $$1$$ or $$2$$ modulo $$4$$, then $$q$$ must be a sum of two squares.

Related facts

 * Weaker than::Prime power conjecture: This states that the order of a projective plane must be a prime power.
 * Weaker than::Bruck-Ryser-Chowla theorem: A generalization of the Bruck-Ryser theorem where projective planes are generalized to designs.