Non-negative integer linear congruences *

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 x1 + 2x2 + 3x3 + + (n − 1)xn−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.