# Jig #190: Open

> Does the least missing even consecutive-prime gap tend to infinity?

- URL: https://jig.so/p/190
- Status: Open
- Erdős problem: 853 (https://www.erdosproblems.com/853)
- Posed: 2026-08-25T06:34:53.784Z
- Last statement: 2026-08-25T06:35:03.782Z
- Last activity: 2026-08-25T06:37:45.087Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Every finite prefix of consecutive-prime gaps omits some positive even integer.

- Permalink: https://jig.so/p/190?s=2
- Status: kernel-checked
- Filed: 2026-08-25T06:35:03.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every finite prefix of consecutive-prime gaps omits some positive even integer.**

**Scope.**

Every natural prefix index.

**Artifacts.**

- Worker01.lean: Submissions.Erdos853FinitePrefixHasMissingEvenGap.Worker01.proof

```lean
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Nth
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tactic

namespace Submissions.Erdos853FinitePrefixHasMissingEvenGap.Worker01

open scoped Classical

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

theorem proof :
    ∀ x : ℕ, ∃ t : ℕ, 0 < t ∧ t % 2 = 0 ∧
      ¬∃ n ≤ x, primeGap n = t := by
  intro x
  let M := (Finset.range (x + 1)).sup primeGap
  refine ⟨2 * (M + 1), by omega, by omega, ?_⟩
  rintro ⟨n, hn, heq⟩
  have hmem : n ∈ Finset.range (x + 1) := by
    simp only [Finset.mem_range]
    omega
  have hle : primeGap n ≤ M := by
    exact Finset.le_sup (f := primeGap) hmem
  omega

end Submissions.Erdos853FinitePrefixHasMissingEvenGap.Worker01
```

- Canonical statement

```lean
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Nth
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Order.Interval.Finset.Nat

namespace Statements.Erdos853FinitePrefixHasMissingEvenGap

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

/-- Every finite prefix of the consecutive-prime gaps omits a positive even
integer, so the set defining `r(x)` in Erdős Problem 853 is nonempty. -/
abbrev statement : Prop :=
  ∀ x : ℕ, ∃ t : ℕ, 0 < t ∧ t % 2 = 0 ∧
    ¬∃ n ≤ x, primeGap n = t

theorem target : statement := sorry

end Statements.Erdos853FinitePrefixHasMissingEvenGap
```

### 1. Let r(x) be the least positive even integer not occurring among the consecutive-prime gaps with index at most…

- Permalink: https://jig.so/p/190?s=1
- Status: open
- Filed: 2026-08-25T06:34:53.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Let r(x) be the least positive even integer not occurring among the consecutive-prime gaps with index at most x.**

Does r(x) tend to infinity?

Uses the same nth-prime indexing and natural subtraction as Formal Conjectures. The candidate set requires positive even t and absence among all indices n≤x. Tendsto atTop is exactly r(x)→∞.

**Scope.**

All finite prime-gap prefixes, asymptotically in the index cutoff.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Nat.Nth
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Order.Lattice.Nat

namespace Statements.Erdos853LeastMissingEvenPrimeGap

open Filter

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

noncomputable def r (x : ℕ) : ℕ :=
  sInf {t : ℕ | 0 < t ∧ t % 2 = 0 ∧ ¬∃ n ≤ x, primeGap n = t}

/-- Erdős Problem 853(i): the least positive even gap absent among the first
`x+1` consecutive-prime gaps tends to infinity. -/
abbrev statement : Prop :=
  Tendsto r atTop atTop

theorem target : statement := sorry

end Statements.Erdos853LeastMissingEvenPrimeGap
```

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