Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-24T03:21:24.240Z Has data issue: false hasContentIssue false
  • English
  • Français

Perfect difference sets

Published online by Cambridge University Press:  18 May 2009

H. Halberstam  and
R. R. Laxton
H. Halberstam
Affiliation:
Trinity College, DublinUniversity of Michigan and University of Sussex
R. R. Laxton
Affiliation:
Trinity College, DublinUniversity of Michigan and University of Sussex
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If the set K of r+1 distinct integers k0, k1 …, kr has the property that the (r+1)r differences ki–kj (0≦i, j≦r, i≠j) are distinct modulo r2+r+1, K is called a perfect difference set modr2+r+1. The existence of perfect difference sets seems intuitively improbable, at any rate for large r, but in 1938 J. Singer [1] proved that, whenever r is a prime power, say r = pn, a perfect difference set mod p2n+pn+1 exists. Since the appearance of Singer's paper several authors have succeeded in showing that for many kinds of number r perfect difference sets mod r2+r+1 do not exist; but it remains an open question whether perfect difference sets exist only when r is a prime power (for a comprehensive survey see [2]).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 6 , Issue 4 , July 1964 , pp. 177 - 184
Copyright
Copyright © Glasgow Mathematical Journal Trust 1964

References

REFERENCES

1.Singer, J., A theorem of finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), 377–385.Google Scholar
2.Hall, M. Jr, A survey of difference sets, Proc. Amer. Math. Soc. 7 (1956), 975–986.CrossRefGoogle Scholar
3.Halberstam, H. and Laxton, R. R., On perfect difference sets, Quart. J. Oxford Ser. (2) 14 (1963), 86–90.Google Scholar
4.Berman, G., Finite projective plane geometries and difference sets, Trans. Amer. Math. Soc. 74 (1953), 492–499.CrossRefGoogle Scholar
5.Bruck, R. H., Difference sets in a finite group. Trans. Amer. Math. Soc. 78 (1955), 464–481.Google Scholar
6.Higman, D. G. and McLaughlin, J. E., Geometric ABA-groups, Illinois J. Math. 5 (1961), 382–397.CrossRefGoogle Scholar
7.Gordon, B., Mills, W. H. and Welch, L. R., Some new difference sets, Canad. J. Math. 14 (1962), 614–625.Google Scholar