2008 | OriginalPaper | Chapter
The Number of Solutions of a Linear Homogeneous Congruence
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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
(1)
$$ \sum\limits_{i = 1}^k {a_i x_i \equiv } 0\left( {\bmod n} \right)with 0 \leqslant x_i \leqslant b_i $$
satisfies the inequality
(2)
$$ N\left( {n;a_1 ,b_1 ; \ldots ;a_k ,b_k } \right) \geqslant 2^{1 - n} \prod\limits_{i = 1}^k {\left( {b_i + 1} \right).} $$