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Some Theorems on Difference Sets

Published online by Cambridge University Press:  20 November 2018

Henry B. Mann
Henry B. Mann*
Affiliation:
Ohio State University
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A set a1 …, ak of different residues mod v is called a difference set (v, k, λ) (v>k > λ) if the congruence ai — aj ≡ d (mod v) has exactly λ solutions for d ≢ 0 (mod v). Singer [4] has demonstrated the existence of a difference set (v, k, 1) if k — 1 is a prime power, and difference sets for λ > 1 have been constructed by various authors; but necessary and sufficient conditions for the existence of a (v, k, λ) are not known. It has not been possible so far to find a difference set with λ = 1 if k — 1 is not a prime power and it has therefore been conjectured that no such difference set exists.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 4 , 1952 , pp. 222 - 226
Copyright
Copyright © Canadian Mathematical Society 1952

References

1. Chowla, S. andRyser, H. J., Combinatorial problems, Can. J. Math., vol. 2 (1950), 93–99.Google Scholar
2. Hall, Marshall Jr., Cyclic projective planes, Duke Math. J., vol. 14 (1947), 1079–1090.Google Scholar
3. Hall, Marshall Jr. and Ryser, H. J., Cyclic incidence matrices, Can. J. Math., vol. 4 (1951), 495–502.Google Scholar
4. Singer, James, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc, vol. 43 (1938), 377–385.Google Scholar