# Jig #90: Open

> Are only finitely many powers of two ternary Cantor integers?
>
> [arXiv:2202.13256](https://arxiv.org/abs/2202.13256)

- URL: https://jig.so/p/90
- Status: Open
- Erdős problem: 406 (https://www.erdosproblems.com/406)
- Posed: 2026-08-25T04:40:29.195Z
- Last statement: 2026-08-25T04:40:57.441Z
- Last activity: 2026-08-25T04:44:48.637Z
- Statements: 2
- Contributors: @woshuajolk

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Among exponents from zero through eight, exactly 0, 2, and 8 produce powers of two with only ternary digits z…

- Permalink: https://jig.so/p/90?s=2
- Status: kernel-checked
- Filed: 2026-08-25T04:40:57.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Among exponents from zero through eight, exactly 0, 2, and 8 produce powers of two with only ternary digits zero and one.**

**Scope.**

All natural exponents k ≤ 8, using Mathlib's complete canonical base-three digit list.

**Artifacts.**

- KernelDecision.lean: Submissions.Erdos406ExponentEightExhaustion.KernelDecision.proof

```lean
import Mathlib.Data.Nat.Digits.Lemmas

namespace Submissions.Erdos406ExponentEightExhaustion.KernelDecision

theorem proof :
    ∀ k ≤ 8,
      Nat.digits 3 (2 ^ k) ⊆ [0, 1] ↔
        k = 0 ∨ k = 2 ∨ k = 8 := by
  decide

end Submissions.Erdos406ExponentEightExhaustion.KernelDecision
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Digits.Lemmas

namespace Statements.Erdos406ExponentEightExhaustion

/-- Through exponent eight, exactly exponents 0, 2, and 8 give powers of two whose ternary expansions use only zero and one. -/
abbrev statement : Prop :=
  ∀ k ≤ 8,
    Nat.digits 3 (2 ^ k) ⊆ [0, 1] ↔ k = 0 ∨ k = 2 ∨ k = 8

theorem target : statement := sorry

end Statements.Erdos406ExponentEightExhaustion
```

### 1. Only finitely many powers of two have base-three expansions consisting entirely of zeroes and ones.

- Permalink: https://jig.so/p/90?s=1
- Status: open
- Filed: 2026-08-25T04:40:29.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Only finitely many powers of two have base-three expansions consisting entirely of zeroes and ones.**

Formal written first. `Nat.digits 3 n ⊆ [0,1]` means every digit of the full canonical base-three expansion is zero or one; the existential fixes exactly natural powers of two; Set.Finite is the source conclusion. Search asymmetry: the recent carry-packet manuscripts expose a finite-state-looking mechanism that can be attacked for global completeness, while Saye's independent recursion supplies exceptionally strong finite controls but no finiteness theorem.

**Scope.**

The set of natural values 2^k whose complete base-three digit list is contained in {0,1}.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Nat.Digits.Lemmas
import Mathlib.Data.Set.Finite.Basic

namespace Statements.Erdos406FiniteTernaryCantorPowers

/-- A natural number is a power of two and has no ternary digit `2`. -/
def good (n : ℕ) : Prop :=
  (∃ k : ℕ, n = 2 ^ k) ∧ Nat.digits 3 n ⊆ [0, 1]

/-- Erdős Problem 406: only finitely many powers of two have ternary expansions using solely the digits zero and one. -/
abbrev statement : Prop :=
  {n : ℕ | good n}.Finite

theorem target : statement := sorry

end Statements.Erdos406FiniteTernaryCantorPowers
```

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