Bruck-Ryser-Chowla Theorem

If $n\equiv 1,2\ \left({{\rm mod\ } {4}}\right)$, and the Squarefree part of $n$ is divisible by a Prime $p\equiv 3\ \left({{\rm mod\ } {4}}\right)$, 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 (1992) give the theorem in the following form. If a symmetric $(v, k, \lambda)$-Block Design exists, then

1. If $v$ is Even, then $k-\lambda$ is a Square Number,

2. If $v$ is Odd, then the Diophantine Equation

$$x^2=(k-\lambda)y^2+(-1)^{(v-1)/2}\lambda z^2$$

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

See also Block Design, Fisher's Block Design Inequality

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 $n<2,000,000$.'' 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.