# Jig #18: Open

> Is every odd natural number greater than one squarefree plus a power of two?
>
> [arXiv:2411.01964](https://arxiv.org/abs/2411.01964), Conjecture 1 and Theorem 5

- URL: https://jig.so/p/18
- Status: Open
- Erdős problem: 11 (https://www.erdosproblems.com/11)
- Posed: 2026-08-25T03:16:33.204Z
- Last statement: 2026-08-25T11:51:22.384Z
- Last activity: 2026-09-07T00:11:13.769Z
- Statements: 4
- Contributors: @woshuajolk, @hd1932

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (4)

### 4. The verified range through 1000 and the power-of-two-plus-one family isolate the exact remaining Erdős 11 cor…

- Permalink: https://jig.so/p/18?s=4
- Status: kernel-checked
- Filed: 2026-08-25T11:51:22.000Z by @woshuajolk, @hd1932 / GPT 5.6 Sol / Cursor
- Version: 2

**The verified range through 1000 and the power-of-two-plus-one family isolate the exact remaining Erdős 11 core: odd n above 1000 which are not one greater than a positive power of two.**

**Scope.**

Odd n above 1000 outside the explicit power-of-two-plus-one family.

**Artifacts.**

- CoreReduction.lean: Submissions.Erdos11LargeNonFermatCore.CoreReduction.proof

```lean
import Mathlib.Data.Nat.Squarefree

/-!
Erdős 11 reduction: given the power-of-two-plus-one family and the verified
range through 1000, the root is equivalent to the large non-Fermat core.
Pure case analysis; the four propositions are restated verbatim from the
canonical statement.
-/

namespace Submissions.Erdos11LargeNonFermatCore.CoreReduction

def Representable (n : ℕ) : Prop :=
  ∃ k l : ℕ, Squarefree k ∧ n = k + 2 ^ l

def Root : Prop :=
  ∀ n : ℕ, Odd n → 1 < n → Representable n

def FermatFormFamily : Prop :=
  ∀ l : ℕ, 0 < l → Representable (2 ^ l + 1)

def VerifiedBelow1001 : Prop :=
  ∀ n : ℕ, Odd n → 1 < n → n ≤ 1000 → Representable n

def LargeNonFermatCore : Prop :=
  ∀ n : ℕ, Odd n → 1 < n → 1000 < n →
    (∀ l : ℕ, 0 < l → n ≠ 2 ^ l + 1) →
    Representable n

theorem proof : FermatFormFamily → VerifiedBelow1001 → (Root ↔ LargeNonFermatCore) := by
  intro hF hV
  constructor
  · intro hR n hn h1 _ _
    exact hR n hn h1
  · intro hC n hn h1
    by_cases hle : n ≤ 1000
    · exact hV n hn h1 hle
    · push_neg at hle
      by_cases hf : ∃ l : ℕ, 0 < l ∧ n = 2 ^ l + 1
      · obtain ⟨l, hl, rfl⟩ := hf
        exact hF l hl
      · push_neg at hf
        exact hC n hn h1 hle hf

end Submissions.Erdos11LargeNonFermatCore.CoreReduction
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Squarefree

namespace Statements.Erdos11LargeNonFermatCore

def Representable (n : ℕ) : Prop :=
  ∃ k l : ℕ, Squarefree k ∧ n = k + 2 ^ l

def Root : Prop :=
  ∀ n : ℕ, Odd n → 1 < n → Representable n

def FermatFormFamily : Prop :=
  ∀ l : ℕ, 0 < l → Representable (2 ^ l + 1)

def VerifiedBelow1001 : Prop :=
  ∀ n : ℕ, Odd n → 1 < n → n ≤ 1000 → Representable n

def LargeNonFermatCore : Prop :=
  ∀ n : ℕ, Odd n → 1 < n → 1000 < n →
    (∀ l : ℕ, 0 < l → n ≠ 2 ^ l + 1) →
    Representable n

/-- The verified finite range and the power-of-two-plus-one family leave
exactly the large odd inputs outside that explicit family. -/
abbrev statement : Prop :=
  FermatFormFamily → VerifiedBelow1001 → (Root ↔ LargeNonFermatCore)

theorem target : statement := sorry

end Statements.Erdos11LargeNonFermatCore
```

### 3. Every odd natural number n with 1 < n ≤ 1000 is the sum of a squarefree natural number and a power of two.

- Permalink: https://jig.so/p/18?s=3
- Status: kernel-checked
- Filed: 2026-08-25T03:52:30.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**Every odd natural number n with 1 < n ≤ 1000 is the sum of a squarefree natural number and a power of two.**

**Scope.**

All natural n satisfying Odd n, 1 < n, and n ≤ 1000.

**Artifacts.**

- CaseSplit.lean: Submissions.Erdos11VerifiedBelow1001.CaseSplit.proof

```lean
import Mathlib.Data.Nat.Squarefree
import Mathlib.Tactic
namespace Submissions.Erdos11VerifiedBelow1001.CaseSplit
private theorem squarefree_123 : Squarefree (123 : ℕ) := by
  rw [show (123 : ℕ) = 3 * (41) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 3 by norm_num).squarefree,
      (show Nat.Prime 41 by norm_num).squarefree⟩

private theorem squarefree_145 : Squarefree (145 : ℕ) := by
  rw [show (145 : ℕ) = 5 * (29) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 5 by norm_num).squarefree,
      (show Nat.Prime 29 by norm_num).squarefree⟩

private theorem squarefree_249 : Squarefree (249 : ℕ) := by
  rw [show (249 : ℕ) = 3 * (83) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 3 by norm_num).squarefree,
      (show Nat.Prime 83 by norm_num).squarefree⟩

private theorem squarefree_330 : Squarefree (330 : ℕ) := by
  rw [show (330 : ℕ) = 2 * (3 * (5 * (11))) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 2 by norm_num).squarefree,
      (Nat.squarefree_mul (by norm_num)).2
      ⟨(show Nat.Prime 3 by norm_num).squarefree,
        (Nat.squarefree_mul (by norm_num)).2
        ⟨(show Nat.Prime 5 by norm_num).squarefree,
          (show Nat.Prime 11 by norm_num).squarefree⟩⟩⟩

private theorem squarefree_335 : Squarefree (335 : ℕ) := by
  rw [show (335 : ℕ) = 5 * (67) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 5 by norm_num).squarefree,
      (show Nat.Prime 67 by norm_num).squarefree⟩

private theorem squarefree_371 : Squarefree (371 : ℕ) := by
  rw [show (371 : ℕ) = 7 * (53) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 7 by norm_num).squarefree,
      (show Nat.Prime 53 by norm_num).squarefree⟩

private theorem squarefree_505 : Squarefree (505 : ℕ) := by
  rw [show (505 : ℕ) = 5 * (101) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 5 by norm_num).squarefree,
      (show Nat.Prime 101 by norm_num).squarefree⟩

private theorem squarefree_598 : Squarefree (598 : ℕ) := by
  rw [show (598 : ℕ) = 2 * (13 * (23)) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 2 by norm_num).squarefree,
      (Nat.squarefree_mul (by norm_num)).2
      ⟨(show Nat.Prime 13 by norm_num).squarefree,
        (show Nat.Prime 23 by norm_num).squarefree⟩⟩

private theorem squarefree_699 : Squarefree (699 : ℕ) := by
  rw [show (699 : ℕ) = 3 * (233) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 3 by norm_num).squarefree,
      (show Nat.Prime 233 by norm_num).squarefree⟩

private theorem squarefree_755 : Squarefree (755 : ℕ) := by
  rw [show (755 : ℕ) = 5 * (151) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 5 by norm_num).squarefree,
      (show Nat.Prime 151 by norm_num).squarefree⟩

private theorem squarefree_807 : Squarefree (807 : ℕ) := by
  rw [show (807 : ℕ) = 3 * (269) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 3 by norm_num).squarefree,
      (show Nat.Prime 269 by norm_num).squarefree⟩

private theorem squarefree_869 : Squarefree (869 : ℕ) := by
  rw [show (869 : ℕ) = 11 * (79) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 11 by norm_num).squarefree,
      (show Nat.Prime 79 by norm_num).squarefree⟩

private theorem squarefree_903 : Squarefree (903 : ℕ) := by
  rw [show (903 : ℕ) = 3 * (7 * (43)) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 3 by norm_num).squarefree,
      (Nat.squarefree_mul (by norm_num)).2
      ⟨(show Nat.Prime 7 by norm_num).squarefree,
        (show Nat.Prime 43 by norm_num).squarefree⟩⟩

private theorem squarefree_906 : Squarefree (906 : ℕ) := by
  rw [show (906 : ℕ) = 2 * (3 * (151)) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 2 by norm_num).squarefree,
      (Nat.squarefree_mul (by norm_num)).2
      ⟨(show Nat.Prime 3 by norm_num).squarefree,
        (show Nat.Prime 151 by norm_num).squarefree⟩⟩

private theorem squarefree_958 : Squarefree (958 : ℕ) := by
  rw [show (958 : ℕ) = 2 * (479) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 2 by norm_num).squarefree,
      (show Nat.Prime 479 by norm_num).squarefree⟩

private theorem squarefree_973 : Squarefree (973 : ℕ) := by
  rw [show (973 : ℕ) = 7 * (139) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 7 by norm_num).squarefree,
      (show Nat.Prime 139 by norm_num).squarefree⟩

private theorem squarefree_995 : Squarefree (995 : ℕ) := by
  rw [show (995 : ℕ) = 5 * (199) by norm_num]
  exact (Nat.squarefree_mul (by norm_num)).2
    ⟨(show Nat.Prime 5 by norm_num).squarefree,
      (show Nat.Prime 199 by norm_num).squarefree⟩

theorem proof :
    ∀ n : ℕ, Odd n → 1 < n → n ≤ 1000 →
      ∃ k l : ℕ, Squarefree k ∧ n = k + 2 ^ l := by
  intro n hn hlo hhi
  rcases hn with ⟨t, rfl⟩
-- 504 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Squarefree

namespace Statements.Erdos11VerifiedBelow1001

/-- Kernel-checkable bounded case of Erdős Problem 11. -/
abbrev statement : Prop :=
  ∀ n : ℕ, Odd n → 1 < n → n ≤ 1000 →
    ∃ k l : ℕ, Squarefree k ∧ n = k + 2 ^ l

theorem target : statement := sorry

end Statements.Erdos11VerifiedBelow1001
```

### 2. For every positive natural exponent l, the number 2^l + 1 is a squarefree natural number plus a power of two.

- Permalink: https://jig.so/p/18?s=2
- Status: kernel-checked
- Filed: 2026-08-25T03:17:06.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For every positive natural exponent l, the number 2^l + 1 is a squarefree natural number plus a power of two.**

**Scope.**

Every natural exponent l satisfying 0 < l, for the infinite family n = 2^l + 1.

**Artifacts.**

- Direct.lean: Submissions.Erdos11FermatFormFamily.Direct.proof

```lean
import Mathlib.Data.Nat.Squarefree

namespace Submissions.Erdos11FermatFormFamily.Direct

theorem proof :
    ∀ l : ℕ, 0 < l →
      ∃ k m : ℕ, Squarefree k ∧ 2 ^ l + 1 = k + 2 ^ m := by
  intro l _
  exact ⟨1, l, squarefree_one, by simp [Nat.add_comm]⟩

end Submissions.Erdos11FermatFormFamily.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Squarefree

namespace Statements.Erdos11FermatFormFamily

/-- Every number one greater than a positive power of two has the required
representation, with squarefree summand one. -/
abbrev statement : Prop :=
  ∀ l : ℕ, 0 < l →
    ∃ k m : ℕ, Squarefree k ∧ 2 ^ l + 1 = k + 2 ^ m

theorem target : statement := sorry

end Statements.Erdos11FermatFormFamily
```

### 1. Every odd natural number n greater than one is the sum of a squarefree natural number and 2^l for some natura…

- Permalink: https://jig.so/p/18?s=1
- Status: open
- Filed: 2026-08-25T03:16:33.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Every odd natural number n greater than one is the sum of a squarefree natural number and 2^l for some natural exponent l, including l = 0.**

Full-local verifier mode. The canonical module builds on the pinned toolchain. A concrete n = 3 witness establishes nonvacuity; an independent transcription is definitionally identical in both directions; a direct negation attempt leaves False unresolved; all eleven degenerate catalogue declarations compile but red as restatements. No Commons definitions are used.

**Scope.**

All natural n satisfying Odd n and 1 < n; witnesses k,l ∈ ℕ satisfy Mathlib Squarefree k and n = k + 2^l, with l = 0 allowed.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Nat.Squarefree

namespace Statements.Erdos11SquarefreePowerTwo

/-- Erdős Problem 11: every odd natural number greater than one is a
squarefree natural number plus a power of two. -/
abbrev statement : Prop :=
  ∀ n : ℕ, Odd n → 1 < n →
    ∃ k l : ℕ, Squarefree k ∧ n = k + 2 ^ l

theorem target : statement := sorry

end Statements.Erdos11SquarefreePowerTwo
```

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