Bruck-Ryser-Chowla Theorem

If , and the Squarefree part of is divisible by a Prime , then no Difference Set of Order exists. Equivalently, if a Projective Plane of order exists, and or 2 (mod 4), then is the sum of two Squares.

Dinitz and Stinson (1992) give the theorem in the following form. If a symmetric -Block Design exists, then

1. If is Even, then is a Square Number,

2. If is Odd, then the Diophantine Equation

has a solution in integers, not all of which are 0.

References

Dinitz, J. H. and Stinson, D. R. A Brief Introduction to Design Theory.'' Ch. 1 in Contemporary Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, pp. 1-12, 1992.

Gordon, D. M. The Prime Power Conjecture is True for .'' Electronic J. Combinatorics 1, R6 1-7, 1994. http://www.combinatorics.org/Volume_1/volume1.html#R6.

Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., 1963.