A Linear Diophantine Problem

Abstract

Let a 1 , a 2 , … , a t be a set of groupwise relatively prime positive integers. Several authors, (2; 3; 5; 6), have determined bounds for the function F(a 1 , …, a t ) defined by the property that the equation 1 has a solution in positive integers X 1 , …, x t for n > F(a 1 , ..., a t ) . If F(a 1 , …, a t ) is a function of this type, it is easy to see that 2 is the corresponding function for the solvability of (1) in non-negative x's. It is well known that a 1 a 2 is the best bound for F(a 1 , a 2 ) and a 1 a 2 — a, 1 — a 2 for G(a 1 , a 2 ). 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(a 1 , a 2 , a 3 ) 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.