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.