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On Residue Difference Sets

Published online by Cambridge University Press:  20 November 2018

Emma Lehmer
Emma Lehmer*
Affiliation:
Pacific Palisades, California
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In recent years the subject of difference sets has attracted a considerable amount of attention in connection with problems in finite geometries [4]. Difference sets arising from higher power residues were first discussed by Chowla [1], who proved that biquadratic residues modulo p form a difference set if (p — l )/4 is an odd square. In this paper we shall prove a similar result for octic residues and develop some necessary conditions which will eliminate all odd power residue difference sets and many others. We also prove that a perfect residue difference set (that is, one in which every difference appears exactly once) contains all the powers of 2 modulo p.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 5 , 1953 , pp. 425 - 432
Copyright
Copyright © Canadian Mathematical Society 1953

References

1. Chowla, S., A property of biquadratic residues, Proc. Nat. Acad. Sci. India, Sec. A, 14 (1944), 45–46.Google Scholar
2. Dickson, L. E., Cydotomy, higher congruences and Waring's problem, Amer. J. Math., 57 (1935), 391–424.Google Scholar
3. Evans, T. A. and Mann, H. B., On simple difference sets, Sankhyā, 2 (1951), 357–364.Google Scholar
4. Marshall Hall, Jr., Cyclic projective planes, Duke Math. J., 14 (1947), 1079–1090.Google Scholar
5. Lehmer, Emma, The quintic character of 2 and 3, Duke Math. J., 18 (1951), 11–18.Google Scholar