Digit $2$ in base $3$ representation of $2^n$

formal-conjectures · arxiv · ams-5 · ams-11

For $n > 8$, $2^n$ is not the the sum of distinct powers of $3$. Expressed here in terms of the base $3$ digits of $n$.

This conjecture is equivalent to the halting of a $15$-state $2$-symbol Turing Machine.

TODO(lezeau): Formalize the Turing Machine version of this problem.

Source: *Hardness of Busy Beaver Value BB(15)*: https://link.springer.com/chapter/10.1007/978-3-031-72621-7_9
This is also https://arxiv.org/abs/2107.12475.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://doi.org/10.2307/2689842
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/2107.12475
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://doi.org/10.1007/978-3-031-72621-7_9
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Arxiv/2107.12475/CollatzLike.lean
Local target: Corpus.Arxiv210712475CollatzLike.CollatzLike
Source SHA-256: 4e7f8e4fb03f273c87dd55e05aa078f53e6b3bb10227d08c6bf38aa4c1b7a1f0
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record