## Abstract

We solve multiple conjectures by Byszewski and Ulas about the base for \({0\leq k \leq N-1}\). The classical construction can only partition

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

*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.## Key words and phrases

Prouhet–Tarry–Escott problem digit sum## Mathematics Subject Classification

primary 11A63 secondary 05A18## Preview

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© Akadémiai Kiadó, Budapest, Hungary 2018