Bruck–Ryser–Chowla theorem

The Bruck–Ryser–Chowla theorem is a result on the combinatorics of symmetric block designs that implies nonexistence of certain kinds of design. It states that if a (v, b, r, k, λ)-design exists with v = b (implying k = r and λ(v − 1) = k(k − 1)), then:

• if v is even, then k − λ is a square;
• if v is odd, then the following Diophantine equation has a nontrivial solution:

  x² − (k − λ)y² − (−1)^(v−1)/2 λ z² = 0.

The theorem was proved in the case of projective planes by Bruck & Ryser (1949). It was extended to symmetric designs by Chowla & Ryser (1950).