Non-negative integer linear congruences

Abstract

We consider the problem of describing all non-negative integer solutions to a linear congruence in many variables. This question may be reduced to solving the congruence x 1 + 2 x 2 + 3 x 3 + + ( n − 1) x n−1 ≡ 0 (mod n) where i ∈ ℕ = {0, 1, 2, }. We consider the monoid of solutions of this equation and prove equivalent two conjectures of Elashvili concerning the structure of these solutions. This yields a simple algorithm for generating most (conjecturally all) of the high degree indecomposablc solutions of the equation.