Notes on the Finiteness of Powers of Two with All Even Digits

Abstract

We study the problem of finding positive integers $n$ such that all the decimal digits of $2^n$ are even, i.e., belong to $\{0,2,4,6,8\}$. Computational checks up to $n=10^{15}$ reveal the known cases $n=1,2,3,6,11$ and no additional instances. We present a self-contained argument, based on a dynamical Borel-Cantelli lemma, that establishes a metric result related to this problem. We show that the set of "initial phases" in a corresponding dynamical system that would generate infinitely many such powers is of Lebesgue measure zero, providing strong probabilistic support for the finiteness conjecture.

This submission replaces the withdrawn v1. The original version contained a critical error in its main proof. This new version corrects the error, replacing the original unsubstantiated claim with a rigorous metric result and clarifying that the specific problem for powers of two is an open conjecture