Non-Negative Integer Linear Congruences

Details Non-Negative Integer Linear Congruences Non-Negative Integer Linear Congruences

2004-09-24

arxiv.org/abs/math/0409489v1

Mathematics

Number Theory

7 pages. This is the write up of some work we did 10 years ago and have not published up until now. It will be submitted for p

Scientific paper

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 $x_1 + 2x_2 + 3x_3 + ... + (n-1)x_{n-1} \equiv 0 \pmod n$ where values of the unknowns, $x_i$, are sought among the non-negative integers. We consider the monoid of solutions of this equation and prove a conjecture of Elashvili concerning the structure of these solutions. This yields a simple algorithm for generating most (conjecturally all) of the high degree indecomposable solutions of the equation.

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