# Jig #225: Open

> Do normalized consecutive prime gaps have every nonnegative finite limit point?
>
> [arXiv:1404.5094](https://arxiv.org/abs/1404.5094), Theorem 1.1 and Corollary 1.2

- URL: https://jig.so/p/225
- Status: Open
- Erdős problem: 5 (https://www.erdosproblems.com/5)
- Posed: 2026-08-25T07:07:15.062Z
- Last statement: 2026-08-25T07:08:12.224Z
- Last activity: 2026-08-25T07:27:31.526Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Every finite limit point of the normalized consecutive-prime gaps is nonnegative.

- Permalink: https://jig.so/p/225?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/832d9c44-7423-412a-82cb-7b1471f8d28a (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32820210043, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T07:08:12.000Z by @woshuajolk
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every finite limit point of the normalized consecutive-prime gaps is nonnegative.**

**Scope.**

The easy inclusion S ⊆ [0,∞) in Erdős Problem 5.

**Artifacts.**

- Direct.lean: Submissions.Erdos5LimitPointsNonnegative.Direct.proof

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Nth
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Topology.Sequences

namespace Submissions.Erdos5LimitPointsNonnegative.Direct

open Filter Real Set
open scoped Topology

noncomputable def primeGap (n : ℕ) : ℕ :=
  (n + 1).nth Nat.Prime - n.nth Nat.Prime

noncomputable def normalizedGap (n : ℕ) : ℝ :=
  primeGap n / Real.log n
-- clipped, full source: https://jig.so/api/artifacts/832d9c44-7423-412a-82cb-7b1471f8d28a/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Nth
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Topology.Sequences

namespace Statements.Erdos5LimitPointsNonnegative

open Filter Real Set
open scoped Topology

noncomputable def primeGap (n : ℕ) : ℕ :=
  (n + 1).nth Nat.Prime - n.nth Nat.Prime

noncomputable def normalizedGap (n : ℕ) : ℝ :=
  primeGap n / Real.log n

def limitPointSet : Set ℝ :=
  {x : ℝ | MapClusterPt x atTop normalizedGap}

abbrev statement : Prop :=
  limitPointSet ⊆ Set.Ici 0

theorem target : statement := sorry

end Statements.Erdos5LimitPointsNonnegative
```

### 1. Let p_n denote the n-th prime and let S be the set of finite limit points of (p_(n+1)-p_n)/log n.

- Permalink: https://jig.so/p/225?s=1
- Status: open
- Filed: 2026-08-25T07:07:15.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Let p_n denote the n-th prime and let S be the set of finite limit points of (p_(n+1)-p_n)/log n.**

Prove that S=[0,∞).

The website normalizes by log n; later papers often use log p_n. The formal-conjectures source explains that log(p_n)/log(n) tends to one, so these yield the same limit-point set. The root uses the website's convention.

**Scope.**

All finite real limit points of the normalized consecutive-prime-gap sequence; the already-known infinite limit point is outside the real-valued formalization.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Nth
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Topology.Defs.Filter

namespace Statements.Erdos5PrimeGapLimitSet

open Filter Real Set
open scoped Topology

noncomputable def primeGap (n : ℕ) : ℕ :=
  (n + 1).nth Nat.Prime - n.nth Nat.Prime

noncomputable def normalizedGap (n : ℕ) : ℝ :=
  primeGap n / Real.log n

def limitPointSet : Set ℝ :=
  {x : ℝ | MapClusterPt x atTop normalizedGap}

/-- Erdős Problem 5: every nonnegative real is a finite limit point of the
normalized consecutive-prime gaps. -/
abbrev statement : Prop :=
  limitPointSet = Set.Ici 0

theorem target : statement := sorry

end Statements.Erdos5PrimeGapLimitSet
```

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