An improvement of Prouhet’s 1851 result on multigrade chains

Abstract

In 1851 Prouhet showed that when N = j k+1 where j and k are positive integers, j ≥ 2, the first N consecutive positive integers can be separated into j sets, each set containing j k integers, such that the sum of the r-th powers of the members of each set is the same for r = 1, 2, . . . , k. In this paper we show that even when N has the much smaller value 2j k , the first N consecutive positive integers can be separated into j sets, each set containing 2j k−1 integers, such that the integers of each set have equal sums of r-th powers for r = 1, 2, . . . , k. Moreover, we show that this can be done in at least {(j − 1)!} k−1 ways. We also show that there are infinitely many other positive integers N = js such that the first N consecutive positive integers can similarly be separated into j sets of integers, each set containing s integers, with equal sums of r-th powers for r = 1, 2, . . . , k, with the value of k depending on the integer N .