Bruck-Ryser theorem

If n congruent 1, 2 (mod 4), and the squarefree part of n is divisible by a prime p congruent 3 (mod 4), then no difference set of order n exists. Equivalently, if a projective plane of order n exists, and n = 1 or 2 (mod 4), then n is the sum of two squares. Dinitz and Stinson give the theorem in the following form. If a symmetric (v, k, λ)-block design exists, then 1. If v is even, then k - λ is a square number, 2. If v is odd, then the Diophantine equation x^2 = (k - λ) y^2 + (-1)^((v - 1)/2) λ z^2 has a solution in integers, not all of which are 0.

block design | difference set | Fisher's block design inequality