Equal sums of like powers and equal products of integers
Equal sums of like powers and equal products of integers
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DOI：
10.1216/rmj-2013-43-3-763
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发表时间：
2013-06
影响因子：
0.8
通讯作者：
A. Choudhry
中科院分区：
文献类型：
--
作者：
A. Choudhry
几位数学家研究了寻找两个不同的整数集 x1, 的问题。 。 。 , xs 和 y1, . 。 。 , ys，使得 Σs i=1 xi = Σs i=1 yk i , k = k1, k2, . 。 。 , kn，其中 ki 是指定的正整数。当 k = 1, 2, . 时的特殊情况。 。 。 , n 是著名的 Tarry-Escott 问题。本文首次详细研究了寻找两个不同的非零整数集合的问题，除了已经提到的条件外，还满足条件 x1x2 · · · xs = y1y2 · · · ys。本文给出了许多此类丢番图系统的参数或数值解，两个例子是方程组 Σ5 i=1 xi = Σ5 i=1 yk i , k = 1, 2, 3, 5 和 ∏5 i=1 xi = ∏5 i=1 yi ，以及方程组 Σ8 i=1 xi = Σ8 i=1 yk i , k 给出的系统= 1, 2, . 。 。 、6、∏8 i=1 xi = ∏8 i=1 yi。还表明，某些具有等幂和等积的丢番图系统没有任何非平凡解。本文末尾提到了一些未解决的问题。
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.