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

TL;DR

This work investigates lower bounds on the base-$b$ digit-sum of powers $a^{n}$, linking growth to the irrationality of $\log d/\log b$, where $d$ is the smallest factor of $a$ with $\gcd(a/d,b)=1$. Employing elementary number-theoretic methods and, in the general case, Baker’s theorem on linear forms in logarithms, the authors obtain strong bounds such as $c_b(a^{n}) > C\log n$ under the irrationality condition, and a weaker bound $c_b(a^{n}) > \frac{\log n}{\log\log n + C}$ unconditionally when the irrationality condition holds. The approach extends to factorials and the least common multiple sequence $\Lambda(n)=\operatorname{lcm}(1,\dots,n)$, yielding $c_b(n!)>C\log n$ (under mild base conditions) and $c_b(\Lambda_n)>C\log\log n$. An expository treatment of the classic result that $c_b(a^{n})\to\infty$ if and only if $\log(a)/\log(b)$ is irrational is provided. Overall, the paper broadens the toolkit for digit-sum analysis across a wide class of sequences using a blend of elementary divisibility tricks and transcendence-theoretic bounds.

Abstract

We prove that the sum of the base-$b$ digits of $a^{n}$ grows at least logarithmically in $n$ if $\log(d)/\log(b)$ is irrational, where $d$ is the smallest factor of $a$ such that $\gcd(a/d, b) = 1$. Our approach uses only elementary number theory and applies to a wide class of sequences, including factorials and $Λ(n) = lcm(1, 2, \ldots, n)$. We conclude with an expository proof of the previously known result that the sum of the base-$b$ digits of $a^{n}$ tends to infinity with $n$ if and only if $\log(a)/\log(b)$ is irrational.

Figures (1)

• Figure 1: Each block contains at least one nonzero digit.

Theorems & Definitions (19)

• Theorem 1
• proof
• Corollary 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof