Settling some sum suppositions

Abstract

We solve multiple conjectures by Byszewski and Ulas about the base b sum-of-digits function. In order to do this, we develop general results about summations over the sum-of-digits function. As a corollary, we describe an unexpected new result about the Prouhet–Tarry–Escott problem. In some cases, this allows us to partition fewer than b^N values into b sets ${\{S_1,\ldots,S_b\}}$ such that

$$\sum_{s\in S_1}s^k = \sum_{s\in S_2}s^k = \cdots = \sum_{s\in S_b}s^k $$

for ${0\leq k \leq N-1}$. The classical construction can only partition b^N values such that the first N powers agree. Our results are amenable to a computational search, which may discover new, smaller solutions to this classical problem.