where fi and gi are homogeneous polynomials with integral coefficients, fi being of degree n and gi being of degree m. If there are no integers s> 1, a k, 3' such that ak = sla , ij = s, where X, g are positive integers such that Xn =,m, then Xk= ak, yij=gi3 is defined to be a primitive solution of (1). If Xk=aQk, yij=fi3 is a primitive solution of (1), then Xk=tXak, Yi =t P3i3 (derived from the primitive solution), where t0 is an integer, X, IA are any positive integers such that Xn=,tm, is also a solution. Two solutions are said to be equivalent if they may be derived from the same primitive solution. Our first theorem concerns the solution of the system'