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 bN 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 bN 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.