The Indecomposable Solutions of Linear Congruences

Abstract

This article considers the minimal non-zero (= indecomposable) solutions of the linear congruence 1for unknown non-negative integers x 1 , . . . , x n , and characterizes the solutions that attain the Eggleton-Erdős bound. Furthermore it discusses the asymptotic behaviour of the number of indecomposable solutions. The results have direct interpretations in terms of zero-sum sequences and invariant theory.A typical problem of additive number theory is the linear congruence: Given m ∈ N 2 and a ∈ Z n , determine x ∈ N n with (A)(Without loss of generality 0 ≤ a i < m for all i.)Note that in this article N stands for the numbers {0, 1, 2, . . .}, and N k for {k, k + 1, . . .}. Think of 0 as being the most natural number.