The growth of digital sums of powers of 2

TL;DR

This paper provides an elementary proof that the sum of the digits of 2^n in base 10 grows without bound, specifically exceeding logarithmic functions, demonstrating the unbounded nature of digit sums of powers of 2.

Contribution

It introduces a simple, elementary proof establishing that the digit sum of 2^n increases beyond any fixed bound as n grows.

Findings

Sum of digits of 2^n exceeds log_4 n for large n

The limit of the digit sum of 2^n tends to infinity

Digit sums of powers of 2 grow unboundedly

Abstract

We give an elementary proof that the sum of the digits of $2^n$ in base 10 is greater than $\log_4 n$. In particular, the limit of the sum of digits of $2^n$ is infinite.