# Jig #230: Open

> Are there infinitely many cluster primes?

- URL: https://jig.so/p/230
- Status: Open
- Erdős problem: 17 (https://www.erdosproblems.com/17)
- Posed: 2026-08-25T07:11:59.077Z
- Last statement: 2026-08-25T07:13:37.095Z
- Last activity: 2026-08-25T07:27:31.499Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The prime 97 is not a cluster prime.

- Permalink: https://jig.so/p/230?s=2
- Status: kernel-checked
- Filed: 2026-08-25T07:13:37.000Z by @woshuajolk
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The prime 97 is not a cluster prime.**

**Scope.**

The first documented non-cluster prime boundary instance.

**Artifacts.**

- Direct.lean: Submissions.Erdos17NinetySevenNotCluster.Direct.proof

```lean
import Mathlib.Algebra.Ring.Parity
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Tactic

namespace Submissions.Erdos17NinetySevenNotCluster.Direct

def IsClusterPrime (p : ℕ) : Prop :=
  p.Prime ∧ 2 < p ∧
    ∀ n : ℕ, Even n → (n : ℤ) ≤ (p : ℤ) - 3 →
      ∃ q₁ q₂ : ℕ,
        q₁.Prime ∧ q₂.Prime ∧ q₁ ≤ p ∧ q₂ ≤ p ∧
          (n : ℤ) = (q₁ : ℤ) - q₂

theorem proof : ¬IsClusterPrime 97 := by
  intro h
  obtain ⟨q₁, q₂, hq₁, hq₂, hq₁le, -, hdiff⟩ :=
    h.2.2 88 (by norm_num) (by norm_num)
  have hq₂two : 2 ≤ q₂ := hq₂.two_le
  have hq₁low : 90 ≤ q₁ := by omega
  interval_cases q₁ <;> norm_num at hq₁
  have : q₂ = 9 := by omega
  subst q₂
  norm_num at hq₂

end Submissions.Erdos17NinetySevenNotCluster.Direct
```

- Canonical statement

```lean
import Mathlib.Algebra.Ring.Parity
import Mathlib.Data.Nat.Prime.Defs

namespace Statements.Erdos17NinetySevenNotCluster

def IsClusterPrime (p : ℕ) : Prop :=
  p.Prime ∧ 2 < p ∧
    ∀ n : ℕ, Even n → (n : ℤ) ≤ (p : ℤ) - 3 →
      ∃ q₁ q₂ : ℕ,
        q₁.Prime ∧ q₂.Prime ∧ q₁ ≤ p ∧ q₂ ≤ p ∧
          (n : ℤ) = (q₁ : ℤ) - q₂

abbrev statement : Prop :=
  ¬IsClusterPrime 97

theorem target : statement := sorry

end Statements.Erdos17NinetySevenNotCluster
```

### 1. A prime p>2 is a cluster prime if every nonnegative even integer n≤p−3 is q₁−q₂ for primes q₁,q₂≤p.

- Permalink: https://jig.so/p/230?s=1
- Status: open
- Filed: 2026-08-25T07:11:59.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**A prime p>2 is a cluster prime if every nonnegative even integer n≤p−3 is q₁−q₂ for primes q₁,q₂≤p.**

Prove that there are infinitely many cluster primes.

The explicit p>2 clause is present in Elsholtz's definition. The formal-conjectures omission changes only whether 2 is called a cluster prime, never whether the set is infinite.

**Scope.**

All natural primes p>2, with nonnegative even differences and both witnessing primes bounded by p.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Algebra.Ring.Parity
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.Set.Finite.Basic

namespace Statements.Erdos17InfiniteClusterPrimes

def IsClusterPrime (p : ℕ) : Prop :=
  p.Prime ∧ 2 < p ∧
    ∀ n : ℕ, Even n → (n : ℤ) ≤ (p : ℤ) - 3 →
      ∃ q₁ q₂ : ℕ,
        q₁.Prime ∧ q₂.Prime ∧ q₁ ≤ p ∧ q₂ ≤ p ∧
          (n : ℤ) = (q₁ : ℤ) - q₂

/-- Erdős Problem 17: there are infinitely many cluster primes. -/
abbrev statement : Prop :=
  Set.Infinite {p : ℕ | IsClusterPrime p}

theorem target : statement := sorry

end Statements.Erdos17InfiniteClusterPrimes
```

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