The Number of Solutions of a Linear Homogeneous Congruence

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 ,b_k) 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))