# Jig #193: Open

> Positive density of six-factorial square witnesses.

- URL: https://jig.so/p/193
- Status: Open
- Erdős problem: 374 (https://www.erdosproblems.com/374)
- Posed: 2026-08-25T06:35:44.050Z
- Last statement: 2026-08-25T06:47:16.302Z
- Last activity: 2026-08-25T06:47:26.276Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The six distinct indices 16,17,30,31,526,527 give a square product of factorials.

- Permalink: https://jig.so/p/193?s=2
- Status: kernel-checked
- Filed: 2026-08-25T06:47:16.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**The six distinct indices 16,17,30,31,526,527 give a square product of factorials.**

**Scope.**

The explicit upper-witness half for the first known D6 candidate 527.

**Artifacts.**

- Direct.lean: Submissions.Erdos374FactorialSquareWitness527.Direct.proof

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Tactic

namespace Submissions.Erdos374FactorialSquareWitness527.Direct

def IsSquare (n : ℕ) : Prop := ∃ q : ℕ, n = q ^ 2

theorem proof :
    IsSquare (Nat.factorial 527 * Nat.factorial 526 *
      Nat.factorial 31 * Nat.factorial 30 *
      Nat.factorial 17 * Nat.factorial 16) := by
  refine ⟨527 * Nat.factorial 526 * Nat.factorial 30 * Nat.factorial 16, ?_⟩
  rw [show Nat.factorial 527 = 527 * Nat.factorial 526 by
    simpa using Nat.factorial_succ 526]
  rw [show Nat.factorial 31 = 31 * Nat.factorial 30 by
    simpa using Nat.factorial_succ 30]
  rw [show Nat.factorial 17 = 17 * Nat.factorial 16 by
    simpa using Nat.factorial_succ 16]
  norm_num [pow_two]

end Submissions.Erdos374FactorialSquareWitness527.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Factorial.Basic

namespace Statements.Erdos374FactorialSquareWitness527

def IsSquare (n : ℕ) : Prop := ∃ q : ℕ, n = q ^ 2

/-- The six distinct factorial indices from the first known `D₆`
candidate `527 = 17 * 31` have square product. -/
abbrev statement : Prop :=
  IsSquare (Nat.factorial 527 * Nat.factorial 526 *
    Nat.factorial 31 * Nat.factorial 30 *
    Nat.factorial 17 * Nat.factorial 16)

theorem target : statement := sorry

end Statements.Erdos374FactorialSquareWitness527
```

### 1. Do the integers m whose smallest representation by a square product of distinct factorials ending at m uses s…

- Permalink: https://jig.so/p/193?s=1
- Status: open
- Filed: 2026-08-25T06:35:44.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Do the integers m whose smallest representation by a square product of distinct factorials ending at m uses six factorials have positive lower density?**

Using a Finset is equivalent to a strictly increasing list a1<⋯<ak=m and enforces distinctness directly.

**Scope.**

The concrete D6 density subquestion.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Nat.Sqrt
import Mathlib.Topology.Instances.Nat

namespace Statements.Erdos374SixFactorialSquareDensity

open Filter Finset
open scoped BigOperators

def IsSquare (n : ℕ) : Prop := ∃ q : ℕ, n = q ^ 2

def HasFactorialSquareWitness (k m : ℕ) : Prop :=
  ∃ A : Finset ℕ,
    A.card = k ∧ m ∈ A ∧
      (∀ a ∈ A, a ≤ m) ∧
        IsSquare (∏ a ∈ A, a.factorial)

def HasMinimalWitnessSize (k m : ℕ) : Prop :=
  2 ≤ k ∧ HasFactorialSquareWitness k m ∧
    ∀ j : ℕ, 2 ≤ j → j < k → ¬HasFactorialSquareWitness j m

noncomputable def countSix (n : ℕ) : ℕ :=
  open scoped Classical in
    ((Finset.Icc 1 n).filter fun m => HasMinimalWitnessSize 6 m).card

/-- Erdős Problem 374's concrete density question:
the integers whose minimal factorial-square witness has size six
have positive lower density. -/
abbrev statement : Prop :=
  ∃ c : ℝ, 0 < c ∧
    ∀ᶠ n in atTop, c * n ≤ countSix n

theorem target : statement := sorry

end Statements.Erdos374SixFactorialSquareDensity
```

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