On the number of solutions to a diophantine equation

Abstract

Let A 1, …, A r, x 1, …, x r , and A be known positive integers. Let f( A) be the number of integer solutions ( x 1, …, x r ) satisfying the Diophantine equation ∑ j=1 r A jx j = A and the conditions 0 ⩽ x i ⩽ x j , j = 1, …, r. This paper expresses f( A) recursively as a linear function of f(0), f(1), …, f( A−1).