Problem #406: Is it true that there are only finitely many powers of $2$...

---

Progress

The only examples seem to be $1$, $4=1+3$, and $256=1+3+3^2+3^5$. If we only allow the digits $1$ and $2$ then $2^{15}$ seems to be the largest such power of $2$.

This would imply via Kummer's theorem that\[3\mid \binom{2^{k+1}}{2^k}\]for all large $k$.

Saye [Sa22] has computed that $2^n$ contains every possible ternary digit for $16\leq n \leq 5.9\times 10^{21}$.

This is mentioned in problem B33 of Guy's collection [Gu04].