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Mathematical Notes

, Volume 79, Issue 3–4, pp 356–365 | Cite as

Combinational properties of sets of residues modulo a prime and the Erdős—Graham problem

  • A. A. Glibichuk
Article

Abstract

Consider an arbitrary ε > 0 and a sufficiently large prime p > 2. It is proved that, for any integer a, there exist pairwise distinct integers x 1, x 2, ..., x N , where N = 8([1/ε + 1/2] + 1)2 such that 1 ≤ x i p ε, i = 1, ..., N, and
$$a \equiv x_1^{ - 1} + \cdots + x_N^{ - 1} (\bmod p)$$
, where x i −1 is the least positive integer satisfying x i −1 x i ≡ 1 (modp). This improves a result of Sparlinski.

Key words

set of residues modulo a prime Erdős—Graham problem symmetric set antisymmetric set Cauchy—Davenport theorem 

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • A. A. Glibichuk
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityRussia

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