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Bruck-Ryser-Chowla Theorem

The Bruck-Ryser-Chowla theorem states that if n=1,2 (mod 4), and the squarefree part of n is divisible by a prime p=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 (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)lambdaz^2

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

See also

Block Design, Difference Set, Fisher's Block Design Inequality

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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<2000000." Elec. J. Combin. 1, No. 1, R6, 1-7, 1994. https://doi.org/10.37236/1186.Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., 1963.

Referenced on Wolfram|Alpha

Bruck-Ryser-Chowla Theorem

Cite this as:

Weisstein, Eric W. "Bruck-Ryser-Chowla Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Bruck-Ryser-ChowlaTheorem.html

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