EQUAL SUMS OF LIKE POWERS AND EQUAL PRODUCTS OF INTEGERS

Several mathematicians have studied the problem of finding two distinct sets of integers x(1), ..., x(s) and y(1), ... , y(s), such that Sigma(s)(i=1) x(i)(k) = Sigma(s)(i=1) y(i)(k), k = k(1), k(2), ... , k(n), where k(i) 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 x(1)x(2) ... x(s) = y(1)y(2) ... y(s). or numerical solutions are given in this paper for many diophantine systems of this type, two examples being the system of equations Sigma(5)(i=1) x(i)(k) = Sigma(5)(i=1) y(i)(k), k = 1, 2, 3, 5, and Pi(5)(i=1) x(i) = Pi(5)(i=1) y(i), and system given by the equations Sigma(8)(i=1) x(i)(k) = Sigma(8)(i=1) y(i)(k), k = 1, 2, ..., 6, and Pi(8)(i=1) x(i) = Pi(8)(i=1) y(i). 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.