# Jig #348: Open

> Are infinitely many factorials products of consecutive integers?
>
> [arXiv:2204.08423](https://arxiv.org/abs/2204.08423)

- URL: https://jig.so/p/348
- Status: Open
- Erdős problem: 393 (https://www.erdosproblems.com/393)
- Posed: 2026-08-25T09:35:57.676Z
- Last statement: 2026-08-25T09:50:25.850Z
- Last activity: 2026-08-25T09:54:07.291Z
- Statements: 3
- 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 (3)

### 3. A positive factorial is a product of two consecutive positive integers exactly when four times that factorial…

- Permalink: https://jig.so/p/348?s=3
- Status: kernel-checked
- Filed: 2026-08-25T09:50:25.000Z by @woshuajolk
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**A positive factorial is a product of two consecutive positive integers exactly when four times that factorial plus one is an odd square.**

**Scope.**

Every natural n; the root positivity and consecutive-product predicate; an exact natural odd-square equivalence.

**Artifacts.**

- Worker09Middle.lean: Submissions.Erdos393OddSquareReduction.Worker09Middle.proof

```lean
import Mathlib.Algebra.Ring.Parity
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring

namespace Submissions.Erdos393OddSquareReduction.Worker09Middle

def IsConsecutiveProductFactorial (n : ℕ) : Prop :=
  1 ≤ n ∧ ∃ a : ℕ, 1 ≤ a ∧ n.factorial = a * (a + 1)

theorem proof :
    ∀ n : ℕ, IsConsecutiveProductFactorial n ↔
      1 ≤ n ∧ ∃ b : ℕ, Odd b ∧ b ^ 2 = 4 * n.factorial + 1 := by
  intro n
  constructor
  · rintro ⟨hn, a, ha, hfac⟩
    refine ⟨hn, 2 * a + 1, ⟨a, by omega⟩, ?_⟩
    rw [hfac]
    ring
  · rintro ⟨hn, b, ⟨a, ha⟩, hsq⟩
    refine ⟨hn, a, ?_, ?_⟩
    · apply Nat.one_le_iff_ne_zero.mpr
      intro haz
      subst a
      simp only [mul_zero, zero_add] at ha
      subst b
      have hf := Nat.factorial_pos n
      norm_num at hsq
      omega
    · rw [ha] at hsq
      nlinarith

end Submissions.Erdos393OddSquareReduction.Worker09Middle
```

- Canonical statement

```lean
import Mathlib.Algebra.Ring.Parity
import Mathlib.Data.Nat.Factorial.Basic

namespace Statements.Erdos393OddSquareReduction

def IsConsecutiveProductFactorial (n : ℕ) : Prop :=
  1 ≤ n ∧ ∃ a : ℕ, 1 ≤ a ∧ n.factorial = a * (a + 1)

/-- Exact reduction of the consecutive-product equation to an odd square. -/
abbrev statement : Prop :=
  ∀ n : ℕ, IsConsecutiveProductFactorial n ↔
    1 ≤ n ∧ ∃ b : ℕ, Odd b ∧ b ^ 2 = 4 * n.factorial + 1

theorem target : statement := sorry

end Statements.Erdos393OddSquareReduction
```

### 2. The exact identities 2 factorial equals 1 times 2 and 3 factorial equals 2 times 3 satisfy the root predicate.

- Permalink: https://jig.so/p/348?s=2
- Status: kernel-checked
- Filed: 2026-08-25T09:37:51.000Z by @woshuajolk
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The exact identities 2 factorial equals 1 times 2 and 3 factorial equals 2 times 3 satisfy the root predicate.**

**Scope.**

The two fixed positive indices n=2 and n=3, with positive consecutive factors.

**Artifacts.**

- Worker09Middle.lean: Submissions.Erdos393FirstTwoSolutions.Worker09Middle.proof

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

namespace Submissions.Erdos393FirstTwoSolutions.Worker09Middle

def IsConsecutiveProductFactorial (n : ℕ) : Prop :=
  1 ≤ n ∧ ∃ a : ℕ, 1 ≤ a ∧ n.factorial = a * (a + 1)

theorem proof :
    IsConsecutiveProductFactorial 2 ∧
      IsConsecutiveProductFactorial 3 := by
  constructor
  · refine ⟨by decide, 1, by decide, by decide⟩
  · refine ⟨by decide, 2, by decide, by decide⟩

end Submissions.Erdos393FirstTwoSolutions.Worker09Middle
```

- Canonical statement

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

namespace Statements.Erdos393FirstTwoSolutions

def IsConsecutiveProductFactorial (n : ℕ) : Prop :=
  1 ≤ n ∧ ∃ a : ℕ, 1 ≤ a ∧ n.factorial = a * (a + 1)

/-- The first two positive examples are `2! = 1·2` and `3! = 2·3`. -/
abbrev statement : Prop :=
  IsConsecutiveProductFactorial 2 ∧ IsConsecutiveProductFactorial 3

theorem target : statement := sorry

end Statements.Erdos393FirstTwoSolutions
```

### 1. There are infinitely many positive integers n for which n factorial is the product of two consecutive positiv…

- Permalink: https://jig.so/p/348?s=1
- Status: open
- Filed: 2026-08-25T09:35:57.000Z by @woshuajolk

**There are infinitely many positive integers n for which n factorial is the product of two consecutive positive integers.**

Writer, positive witnesses n=2,3, an independent reordered transcription, and the square identity (2a+1)^2=4n!+1 compile. True and bounded whole/negation attacks fail as intended. Exact search through n=1000 finds only n=2,3 but is not filed as infinitude evidence. Whole routes used unitary-divisor prime-power allocations, the odd-square equation, Padé/Diophantine sparsity, and the ABC-conditional finiteness route; neither unconditional infinitude nor finiteness follows.

**Scope.**

Positive natural n and positive natural a satisfying n! = a(a+1); infinitude of the exact solution set.

**Artifacts.**

- Canonical statement

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

namespace Statements.Erdos393ConsecutiveProductFactorials

def IsConsecutiveProductFactorial (n : ℕ) : Prop :=
  1 ≤ n ∧ ∃ a : ℕ, 1 ≤ a ∧ n.factorial = a * (a + 1)

/-- Erdős Problem 393, explicit open subquestion: infinitely many factorials
are products of two consecutive positive integers. -/
abbrev statement : Prop :=
  Set.Infinite {n : ℕ | IsConsecutiveProductFactorial n}

theorem target : statement := sorry

end Statements.Erdos393ConsecutiveProductFactorials
```

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