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Several mathematicians have studied the problem of finding two distinct sets of integers x1, . . . , xs and y1, . . . , ys, such that ∑s i=1 xi = ∑s i=1 yk i , k = k1, k2, . . . , kn, where ki are specified positive integers. The particular case when k = 1, 2, . . . , n is the well-known Tarry-Escott problem. This paper is the first detailed study of the problem of finding two distinct sets of nonzero integers which, in addition to the conditions already mentioned, also satisfy the condition x1x2 · · · xs = y1y2 · · · ys. Parametric or numerical solutions are given in this paper for many diophantine systems of this type, two examples being the system of equations ∑5 i=1 xi = ∑5 i=1 yk i , k = 1, 2, 3, 5, and ∏5 i=1 xi = ∏5 i=1 yi, and the system given by the equations ∑8 i=1 xi = ∑8 i=1 yk i , k = 1, 2, . . . , 6, and ∏8 i=1 xi = ∏8 i=1 yi. It is also shown that certain diophantine systems with equal sums of powers and equal products do not have any nontrivial solutions. Some open problems are mentioned at the end of the paper.

Equal sums of like powers and equal products of integers