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Some Theorems on Difference Sets

  • Henry B. Mann (a1)

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A set a 1 …, a k of different residues mod v is called a difference set (v, k, λ) (v > k > λ) if the congruence a i — ajd (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.

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References

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1. Chowla, S. andRyser, H. J., Combinatorial problems, Can. J. Math., vol. 2 (1950), 9399.
2. Hall, Marshall Jr., Cyclic projective planes, Duke Math. J., vol. 14 (1947), 10791090.
3. Hall, Marshall Jr. and Ryser, H. J., Cyclic incidence matrices, Can. J. Math., vol. 4 (1951), 495502.
4. Singer, James, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc, vol. 43 (1938), 377385.
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