OPEN
This is open, and cannot be resolved with a finite computation.
Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$?
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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].
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This page was last edited 30 September 2025. View history
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Formalised statement?
Yes
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Likes this problem
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chamb,
Dogmachine
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Interested in collaborating
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chamb
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Currently working on this problem
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chamb
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This problem looks difficult
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Dogmachine
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This problem looks tractable
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chamb
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The results on this problem could be formalisable
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None
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I am working on formalising the results on this problem
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None
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Additional thanks to: Desmond Weisenberg
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #406, https://www.erdosproblems.com/406, accessed 2026-04-11
