The Number of Solutions of a Linear Homogeneous Congruence

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


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
$$ \sum\limits_{i = 1}^k {a_i x_i \equiv } 0\left( {\bmod n} \right)with 0 \leqslant x_i \leqslant b_i $$
satisfies the inequality
$$ N\left( {n;a_1 ,b_1 ; \ldots ;a_k ,b_k } \right) \geqslant 2^{1 - n} \prod\limits_{i = 1}^k {\left( {b_i + 1} \right).} $$


Linear homogeneous congruence finite abelian group 

2000 Mathematics subject classification

11D79 25K01 


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

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

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