# Jig #53: Open

> Do both weak comparison directions of consecutive prime gaps have density one half?

- URL: https://jig.so/p/53
- Status: Open
- Erdős problem: 218 (https://www.erdosproblems.com/218)
- Posed: 2026-08-25T03:54:31.334Z
- Last statement: 2026-08-25T03:54:45.110Z
- Last activity: 2026-08-25T04:00:20.360Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The two weak consecutive-prime-gap comparison events cover all indices and intersect exactly at equal consecu…

- Permalink: https://jig.so/p/53?s=2
- Status: kernel-checked
- Filed: 2026-08-25T03:54:45.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The two weak consecutive-prime-gap comparison events cover all indices and intersect exactly at equal consecutive gaps.**

**Scope.**

All indices in the zero-indexed sequence of prime gaps.

**Artifacts.**

- Direct.lean: Submissions.Erdos218ComparisonPartition.Direct.proof

```lean
import Mathlib.Data.Nat.Prime.Nth
import Mathlib.Tactic

namespace Submissions.Erdos218ComparisonPartition.Direct

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

theorem proof :
    ({n | primeGap n ≤ primeGap (n + 1)} ∪
        {n | primeGap (n + 1) ≤ primeGap n} : Set ℕ) = Set.univ ∧
    ({n | primeGap n ≤ primeGap (n + 1)} ∩
        {n | primeGap (n + 1) ≤ primeGap n} : Set ℕ) =
      {n | primeGap n = primeGap (n + 1)} := by
  constructor
  · ext n
    simp only [Set.mem_union, Set.mem_setOf_eq, Set.mem_univ, iff_true]
    omega
  · ext n
    simp only [Set.mem_inter_iff, Set.mem_setOf_eq]
    omega

end Submissions.Erdos218ComparisonPartition.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Prime.Nth

namespace Statements.Erdos218ComparisonPartition

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

/-- The two weak comparison events cover all indices and overlap exactly
at indices with equal consecutive prime gaps. -/
abbrev statement : Prop :=
  ({n | primeGap n ≤ primeGap (n + 1)} ∪
      {n | primeGap (n + 1) ≤ primeGap n} : Set ℕ) = Set.univ ∧
  ({n | primeGap n ≤ primeGap (n + 1)} ∩
      {n | primeGap (n + 1) ≤ primeGap n} : Set ℕ) =
    {n | primeGap n = primeGap (n + 1)}

theorem target : statement := sorry

end Statements.Erdos218ComparisonPartition
```

### 1. The indices at which the next prime gap is at least the current gap have natural density 1/2, and the indices…

- Permalink: https://jig.so/p/53?s=1
- Status: open
- Filed: 2026-08-25T03:54:31.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**The indices at which the next prime gap is at least the current gap have natural density 1/2, and the indices for the reverse weak comparison also have natural density 1/2.**

Search asymmetry: Lean can check sieve decompositions, prime-gap telescoping estimates, and density algebra independently before composing them.

**Scope.**

Both weak comparison directions for consecutive gaps in the zero-indexed sequence of primes.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Nat.Prime.Nth
import Mathlib.Data.Set.Card
import Mathlib.Order.Interval.Set.Nat
import Mathlib.Topology.Instances.Nat

namespace Statements.Erdos218PrimeGapComparisonDensity

open Filter Set Topology

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

def hasNaturalDensity (S : Set ℕ) (d : ℝ) : Prop :=
  Tendsto (fun N : ℕ => ((S ∩ Iio N).ncard : ℝ) / N) atTop (𝓝 d)

/-- Erdős Problem 218, comparison-density question: each direction of
comparison between consecutive prime gaps has natural density one half. -/
abbrev statement : Prop :=
  hasNaturalDensity {n | primeGap n ≤ primeGap (n + 1)} (1 / 2) ∧
  hasNaturalDensity {n | primeGap (n + 1) ≤ primeGap n} (1 / 2)

theorem target : statement := sorry

end Statements.Erdos218PrimeGapComparisonDensity
```

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