Note on linear forms

Abstract

where the ai are fixed positive integers with gcd = 1 and the xi are non-negative integers. In particular the question of finding the smallest n for which all greater integers have a solution has been investigated to some extent [1; 2]. It seems that the solution for s = 1 has been known for some time but that the problem in general remains unsolved for s> 1. In the paper of A. Brauer cited in the bibliography various upper bounds for the smallest n are given and the actual value of the smallest n is determined for the ai consecutive integers. The main result of this paper is the determination of this smallest n when the ai are in arithmetical progression. 2. Our investigation then is with the linear form