# Jig #148: Open

> Are there finitely many large least-prime-factor binomial exceptions?

- URL: https://jig.so/p/148
- Status: Open
- Erdős problem: 1094 (https://www.erdosproblems.com/1094)
- Posed: 2026-08-25T06:00:23.877Z
- Last statement: 2026-08-25T06:02:35.153Z
- Last activity: 2026-08-25T06:12:58.234Z
- 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. There are no exceptions to the Problem 1094 least-prime-factor bound when k=1.

- Permalink: https://jig.so/p/148?s=2
- Status: kernel-checked
- Filed: 2026-08-25T06:02:35.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**There are no exceptions to the Problem 1094 least-prime-factor bound when k=1.**

**Scope.**

The complete k=1 slice for all natural n.

**Artifacts.**

- Direct.lean: Submissions.Erdos1094NoWidthOneExceptions.Direct.proof

```lean
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic

namespace Submissions.Erdos1094NoWidthOneExceptions.Direct

theorem proof : ∀ n : ℕ,
    ¬(0 < 1 ∧ 2 * 1 ≤ n ∧
      (n.choose 1).minFac > max (n / 1) 1) := by
  intro n h
  have hn : 0 < n := by omega
  have hmin := Nat.minFac_le hn
  simp only [Nat.choose_one_right, Nat.div_one] at h
  have hmax : max n 1 = n := max_eq_left (by omega)
  rw [hmax] at h
  omega

end Submissions.Erdos1094NoWidthOneExceptions.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Factorization.Basic

namespace Statements.Erdos1094NoWidthOneExceptions

/-- The complete `k=1` slice of Erdős Problem 1094 has no exceptions. -/
abbrev statement : Prop :=
  ∀ n : ℕ,
    ¬(0 < 1 ∧ 2 * 1 ≤ n ∧
      (n.choose 1).minFac > max (n / 1) 1)

theorem target : statement := sorry

end Statements.Erdos1094NoWidthOneExceptions
```

### 1. Among pairs n≥2k>0, are there only finitely many for which the least prime factor of C(n,k) exceeds max(n/k,k…

- Permalink: https://jig.so/p/148?s=1
- Status: open
- Filed: 2026-08-25T06:00:23.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Among pairs n≥2k>0, are there only finitely many for which the least prime factor of C(n,k) exceeds max(n/k,k)?**

Nat.minFac follows Mathlib's convention; admissibility ensures the binomial coefficient is positive and nontrivial where needed.

**Scope.**

All admissible natural pairs (n,k).

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Set.Finite.Basic

namespace Statements.Erdos1094BinomialLeastPrimeExceptions

/-- Erdős Problem 1094: only finitely many admissible binomial coefficients
have least prime factor larger than both `n/k` and `k`. -/
abbrev statement : Prop :=
  {(n, k) : ℕ × ℕ |
      0 < k ∧ 2 * k ≤ n ∧
        (n.choose k).minFac > max (n / k) k}.Finite

theorem target : statement := sorry

end Statements.Erdos1094BinomialLeastPrimeExceptions
```

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