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

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.