Problem 406

Erdős Problem #406: declared status 'open'. Formalized: yes. Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$? Current best: Saye [Sa22] has computed that $2^n$ contains every possible ternary digit for $16\leq n \leq 5.9\times 10^{21}$. Prize: no. Tags: base representations, number theory.