Elementary Bounds on Digital Sums of Powers, Factorials, and LCMs

TL;DR

This paper establishes elementary logarithmic lower bounds on digital sums of powers, factorials, and LCMs, and provides an expository proof of Stewart's theorem utilizing Baker's theorem on logarithms.

Contribution

It introduces elementary methods to derive lower bounds on digital sums and offers an accessible proof of Stewart's theorem using advanced logarithmic techniques.

Findings

Logarithmic lower bounds for digital sums of powers and factorials

Elementary proof techniques for bounds on digital sums

Expository proof of Stewart's theorem using Baker's theorem

Abstract

We prove logarithmic lower bounds on digital sums of powers, multiples of powers, factorials, and the least common multiple of $\{1,\ldots, n\}$, using only elementary number theory. We conclude with an expository proof of Stewart's theorem on digital sums of powers, which uses Baker's theorem on linear forms in logarithms.