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The Number of Solutions of a Linear Homogeneous Congruence

  • Andrzej Schinzel
Part of the Developments in Mathematics book series (DEVM, volume 16)

Abstract

The aim of this paper is to propose and to study the following Conjecture. Let n ∈ ℕ, a i ∈ ℤ and b i ∈ ℕ(l ≤ i ≤ k). The number N(n; a 1 , b 1 ; ; a k ,bk) of solutions of the congruence
$$ \sum\limits_{i = 1}^k {a_i x_i \equiv } 0\left( {\bmod n} \right)with 0 \leqslant x_i \leqslant b_i $$
(1)
satisfies the inequality
$$ N\left( {n;a_1 ,b_1 ; \ldots ;a_k ,b_k } \right) \geqslant 2^{1 - n} \prod\limits_{i = 1}^k {\left( {b_i + 1} \right).} $$
(2)

Keywords

Linear homogeneous congruence finite abelian group 

2000 Mathematics subject classification

11D79 25K01 

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References

  1. 1.
    Drmota, M., Skalba, M.: Equidistribution of divisors in residue classes. Preprint. Technical University of Vienna, ViennaGoogle Scholar
  2. 2.
    Korobov, N.M.: Teoretikochislovye Metody v Priblizhennom Analise. Gos. Izd’vo Fiziko-matematiche-skoi Lit’ry, Moscow (1963)Google Scholar
  3. 3.
    Mann, H.B.: Addition Theorems. The Addition Theorems of Group Theory and Number Theory. Interscience, New York (1965)zbMATHGoogle Scholar
  4. 4.
    Olson, J.: A combinatorial problem in finite abelian groups, II. J. Number Theory 1, 195–199 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Schinzel, A.: The number of solutions of a linear homogeneous congruence II. In: Chen, W., Gowers, T., Halberstam, H., Schmidt, W., Vaughan, R.C. (eds.) Analytical Number Theory: Essays in Honour of Klaus F. Roth. Cambridge University Press, Cambridge (to appear)Google Scholar
  6. 6.
    Schinzel, A., Zakarczemny, M.: On a linear homogeneous congruence. Colloq. Math. 106, 283–292 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Zakarczemny, M.: Master dissertation, Department of Mathematics, University of Warsaw, Warsaw, Poland (2004)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Andrzej Schinzel
    • 1
  1. 1.Instytut MatematycznyPolskiej Akademii NaukWarsawPoland

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