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

Abstract

We prove logarithmic lower bounds on digital sums of powers, multiples of powers, factorials, and the least common multiple of $\{1,\dots ,n\}$, 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.

arXiv comment15 pages, 3 figures. This version corrects errors in the earlier manuscript and substantially clarifies the arguments by adding detailed explanations and additional figures