Settling some sum suppositions

TL;DR

This paper proves several conjectures about the sum of digits function in various bases, introduces new results related to the Prouhet-Tarry-Escott problem, and enables smaller partitions with equal power sums.

Contribution

It develops general summation results for the sum of digits function and applies them to find smaller partitions solving the Prouhet-Tarry-Escott problem.

Findings

Resolved multiple conjectures about sum of digits functions.

Discovered new partitions with fewer than b^N values for equal power sums.

Enabled computational search for smaller solutions to classical problems.

Abstract

We solve multiple conjectures by Byszewski and Ulas about the sum of base $b$ 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.