A Linear Diophantine Problem

Abstract

Let a1, a2, … , at be a set of groupwise relatively prime positive integers. Several authors, (2; 3; 5; 6), have determined bounds for the function F(a1, …, at) defined by the property that the equation1has a solution in positive integers X1, …, xt for n > F(a1, ..., at). If F(a1, …, at) is a function of this type, it is easy to see that2is the corresponding function for the solvability of (1) in non-negative x's.It is well known that a1a2 is the best bound for F(a1, a2) and a1a2 — a,1 — a2 for G(a1, a2). Otherwise only in very special cases have the best bounds been found, even for t = 3.In the present paper a symmetric expression is developed for the best bound for F(a1, a2, a3) which solves that problem and gives insight on the general problem for larger values of t. In addition, some relations are developed which may be of interest in themselves.