Bruck-Ryser theorem

Statement

If there exists a Projective plane (?) of finite order $q$ (or equivalently, there exists an 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

• Prime power conjecture: This states that the order of a projective plane must be a prime power. For full proof, refer: Prime power conjecture implies Bruck-Ryser theorem
• Bruck-Ryser-Chowla theorem: A generalization of the Bruck-Ryser theorem where projective planes are generalized to designs.