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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
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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

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