In 1851, Prouhet showed that when [Formula: see text] where [Formula: see text] and [Formula: see text] are positive integers, [Formula: see text], the first [Formula: see text] consecutive positive integers can be separated into [Formula: see text] sets, each set containing [Formula: see text] integers, such that the sum of the [Formula: see text]th powers of the members of each set is the same for [Formula: see text]. In this paper, we show that even when [Formula: see text] has the much smaller value [Formula: see text], the first [Formula: see text] consecutive positive integers can be separated into [Formula: see text] sets, each set containing [Formula: see text] integers, such that the integers of each set have equal sums of [Formula: see text]th powers for [Formula: see text]. Moreover, we show that this can be done in at least [Formula: see text] ways. We also show that there are infinitely many other positive integers [Formula: see text] such that the first [Formula: see text] consecutive positive integers can similarly be separated into [Formula: see text] sets of integers, each set containing [Formula: see text] integers, with equal sums of [Formula: see text]th powers for [Formula: see text], with the value of [Formula: see text] depending on the integer [Formula: see text].

中文翻译：

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Prouhet 1851 年多级链结果的改进

1851年，Prouhet证明当[公式：见正文]其中[公式：见正文]和[公式：见正文]为正整数时，[公式：见正文]，第一个[公式：见正文]连续的正整数可以被分成 [Formula: see text] 集合，每个集合包含 [Formula: see text] 整数，使得每个集合的成员的 [Formula: see text] 次方之和对于 [Formula: see text] 相同文本]。在本文中，我们表明，即使 [公式：参见文本] 具有小得多的值 [公式：参见文本]，第一个 [公式：参见文本] 连续的正整数也可以分成 [公式：参见文本] 集合，每个集合包含 [公式：参见文本] 整数，使得每个集合的整数具有相等的和 [公式：参见文本] 的 [公式：参见文本] 次方。此外，我们证明这至少可以在 [公式：见文]方式。我们还证明了还有无限多的其他正整数 [公式：参见文本]，因此第一个 [公式：参见文本] 连续的正整数可以类似地分成 [公式：参见文本] 整数集，每个集合包含 [公式：见正文]整数，[公式：见正文]的[公式：见正文]次方之和相等，[公式：见正文]的值取决于整数[公式：见正文]。