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.

© 1996-9

1999-05-26