EQUAL SUMS OF LIKE POWERS, BOTH POSITIVE AND NEGATIVE

摘要

Several mathematicians have studied the problem of finding two sets of integers x_1... x_s and y_1, ... y_s, such that Σ_(i=1)~s.x_i~r = Σ_(i=1)y_i~r,r=k_1, k_2...,k_n, where k_i are specified positive integers. The particular case when r = 1, 2, ... , n is the well-known Tarry-Escott problem. This paper is concerned with the scarcely investigated problem of finding two or more distinct sets of integers with equal sums of powers for both positive and negative powers, that is to say, integer solutions of diophantine systems Σ_(i=1)~sx_i~r =Σ_(i=1)~sy_i~r and diophantine chains Σ_(i=1)~sx_(il)~r =Σ_(i=1)~sx_(i2)~r =...= where, in both cases, the equality holds simulta- neously for negative integral exponents,-h_m,...-h_2 and positive integral exponents k_1, k_2,..., k_n. It is proved in the paper that, given any arbitrary set of such exponents, there exists a solution of the aforementioned diophantine chains for a suitable value of s. Parametric or numerical solutions of many diophantine systems and chains are given in the paper, two examples being the system of equations Σ_(i=1)~9 x_(il)~r=Σ_(i=1)~9x_(i2)~r= -2, -1,1,3, 5, 7. trarily long chains Σ_(i=1)~6 x_(il)~r=Σ_(i=1)~6 x_(i2)~r=Σ_(i=1)~6 x_(it)~r, r=-1,1,2,3,4,5.