Erdős Problem #406

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

Write $2^k=\sum_{j=0}^{J}d_j3^j$ in its finite base-three expansion, where $d_j\in\{0,1,2\}$ and $d_J\ne0$. The admissible exponents $k\geq0$ are those for which no digit is two. The question asks whether the set of admissible exponents is finite. [S1] [S2]

Statement recorded in the supplied source.

Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$? [S1]

Short English statement

Are there only finitely many powers of two whose base-three expansions avoid the digit two?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2098 (EP-406), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 406, https://www.erdosproblems.com/406. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 R. I. Saye, On two conjectures concerning the ternary digits of powers of two, J. Integer Seq. 25 (2022), Article 22.3.4. https://arxiv.org/abs/2202.13256. Bibliographic information from the supplied record.

• S4 J. C. Lagarias, Ternary expansions of powers of 2, J. London Math. Soc. 79 (2009), 562-588. https://arxiv.org/abs/math/0512006. Bibliographic information from the supplied record.

• S5 Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. Bibliographic information from the supplied record.