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Updated Oct 7

Problem 406

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

Status. Open.

Source. erdosproblems.com/406, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #406, https://www.erdosproblems.com/406.

References.

[AbLa14] Abram, William C. and Lagarias, Jeffrey C., Intersections of multiplicative translates of 3-adic Cantor sets. J. Fractal Geom. 1 (2014), no. 4, 349-390; doi:10.4171/JFG/11.
[Gu04] Guy, Richard K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section B33 "Largest divisor of a binomial coefficient", printed p. 135: "Erdős has also conjectured that for $k>8$ , $2^k$ is not the sum of distinct powers of 3 [ $2^8=3^5+3^2+3+1$ ]", a stronger form of the page's question (no such power beyond $2^8$ , where the page asks only for finitely many), stated for its consequence that $3\mid\binom{2^{k+1}}{2^k}$ for $k\ge9$ . Library home: guy_2004_unsolved_problems_number_theory.
[La09] Lagarias, Jeffrey C., Ternary expansions of powers of 2. J. Lond. Math. Soc. (2) (2009), 562-588.
[Na80] Narkiewicz, W., A note on a paper of H. Gupta concerning powers of two and three: ``Powers of $2$ \ and sums of distinct powers of $3$ ''\ [Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 602-633 (1978), 151-158 (1979);\ MR 81g:10016]. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. (1980), 173-174.
[Sa22] Saye, Robert I., On two conjectures concerning the ternary digits of powers of two. J. Integer Seq. (2022), Art. 22.3.4, 9.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

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Linked from (20)

Diophantine Problems and Powers Diophantine Problems and Powers diophantine_problems/abram_2014_intersections_multiplicative_translates_3_adic_cantor Conjecture 1.2 (p. 4): the 3-adic exceptional set has Hausdorff dimension zero Theorem 1.6 (p. 7): dim_H C(1, M_1, ..., M_n) = log_3 beta for a Perron eigenvalue beta Theorem 1.8 (p. 8): dim_H C(1, 3^k + 1) = log_3((1 + sqrt 5)/2)Theorem 1.9 (p. 8): dim_H of the generalized exceptional set is at least (1/2) log_3 2 Theorem 5.2 (p. 30): lower bounds for dim_H E^(2)(Z_3) and dim_H E^(3)(Z_3)diophantine_problems/lagarias_2009_ternary_expansions_powers_2 Theorem 1.1 (p. 2): at most 25 X^0.9725 truncated doublings floor(lambda 2^n) omit the digit 2 Theorem 1.2 (p. 3): uncountably many lambda > 0 with floor(lambda 2^n) omitting the digit 2 along a sparse infinite set of n Theorem 1.3 (p. 3): the truncated real exceptional set has Hausdorff dimension log_3 2 Theorem 1.4 (p. 4): at most 2 X^(log_3 2) of the 3-adic doublings lambda 2^n omit the digit 2 Theorem 1.5 (pp. 4--5): Hausdorff dimensions of the 3-adic sets E^(1), E^(2), E^(3)Theorem 1.6 (p. 6): dim_H C(1, M) <= 1/2 when M is not a power of 3 Theorem 1.7 (p. 6): a lower bound for dim_H C(M_1, ..., M_k) from one integer N diophantine_problems/saye_2022_two_conjectures_concerning_ternary_digits_powers Lemma 1 (p. 3): the order of 2 modulo 3^k and how the (k+1)st ternary digit of 2^n moves Main result (pp. 1, 5, unnumbered): computer check of the Erdős and Sloane ternary-digit conjectures up to 2·3^45 number_theory/guy_2004_unsolved_problems_number_theory

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