An improvement of Prouhet’s 1851 result on multigrade chains

Abstract

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].