Bruck-Ryser-Chowla Theorem
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
has a solution in integers, not all of which are 0.
See also
Block Design, Difference Set, Fisher's Block Design InequalityExplore with Wolfram|Alpha
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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
." Electronic J. Combinatorics 1, No. 1, R6, 1-7, 1994. http://www.combinatorics.org/Volume_1/Abstracts/v1i1r6.html.Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., 1963.
Referenced on Wolfram|Alpha
Bruck-Ryser-Chowla TheoremCite this as:
Weisstein, Eric W. "Bruck-Ryser-Chowla Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Bruck-Ryser-ChowlaTheorem.html