# Jig #213: Open

> Are longest runs of distinct consecutive prime gaps o(log x)?

- URL: https://jig.so/p/213
- Status: Open
- Erdős problem: 852 (https://www.erdosproblems.com/852)
- Posed: 2026-08-25T06:56:49.149Z
- Last statement: 2026-08-25T07:14:37.908Z
- Last activity: 2026-08-25T07:14:57.903Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. A pairwise-distinct run of consecutive prime gaps, all at most B, has length at most B+1.

- Permalink: https://jig.so/p/213?s=2
- Status: kernel-checked
- Filed: 2026-08-25T07:14:37.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**A pairwise-distinct run of consecutive prime gaps, all at most B, has length at most B+1.**

**Scope.**

Every finite block of consecutive prime gaps and every uniform natural upper bound on the gaps in that block.

**Artifacts.**

- Pigeonhole.lean: Submissions.Erdos852BoundedDistinctRun.Pigeonhole.proof

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

namespace Submissions.Erdos852BoundedDistinctRun.Pigeonhole

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

def DistinctRun (start length : ℕ) : Prop :=
  (Finset.range length : Set ℕ).Pairwise fun i j =>
    primeGap (start + i) ≠ primeGap (start + j)

theorem proof :
    ∀ start length B : ℕ,
      DistinctRun start length →
        (∀ i < length, primeGap (start + i) ≤ B) →
          length ≤ B + 1 := by
  intro start length B hdistinct hbounded
  let f : Fin length → Fin (B + 1) := fun i =>
    ⟨primeGap (start + i), by
      exact Nat.lt_succ_of_le (hbounded i i.isLt)⟩
  have hf : Function.Injective f := by
    intro i j hij
    apply Fin.ext
    by_contra hne
    have hgap :
        primeGap (start + (i : ℕ)) ≠
          primeGap (start + (j : ℕ)) := by
      exact hdistinct
        (Finset.mem_range.mpr i.isLt)
        (Finset.mem_range.mpr j.isLt)
        hne
    exact hgap (congrArg Fin.val hij)
  have hcard := Fintype.card_le_of_injective f hf
  simpa using hcard

end Submissions.Erdos852BoundedDistinctRun.Pigeonhole
```

- Canonical statement

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

/-!
# A finite obstruction for Erdős problem 852

A run of pairwise distinct prime gaps all bounded by `B` has length at most
`B + 1`, by the pigeonhole principle.
-/

namespace Statements.Erdos852BoundedDistinctRun

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

def DistinctRun (start length : ℕ) : Prop :=
  (Finset.range length : Set ℕ).Pairwise fun i j =>
    primeGap (start + i) ≠ primeGap (start + j)

abbrev statement : Prop :=
  ∀ start length B : ℕ,
    DistinctRun start length →
      (∀ i < length, primeGap (start + i) ≤ B) →
        length ≤ B + 1

theorem target : statement := sorry

end Statements.Erdos852BoundedDistinctRun
```

### 1. The longest pairwise-distinct run of consecutive prime gaps starting before index x is o(log x).

- Permalink: https://jig.so/p/213?s=1
- Status: open
- Filed: 2026-08-25T06:56:49.000Z by @woshuajolk

**The longest pairwise-distinct run of consecutive prime gaps starting before index x is o(log x).**

Six-role fleet passed: independent nth-prime gap, pairwise-run, and supremum definitions agree both ways; kernel checks give the first gaps 1 and 2, the initial two-gap run is distinct, and every one-gap run is distinct; false-premise control preflights red/restatement; negation remains the exact open little-o claim; final Jig dedupe found only distinct prime-gap problems. Whole attacks used the quadratic minimum sum of distinct even gaps, global second-moment gap bounds, local sieve repetition, Cramer-model birthday heuristics, and prescribed prime-tuple refutation templates. Known arguments give only polynomial upper bounds; fixed-length prime-tuple theorems cannot construct logarithmically growing distinct runs. No full settlement or new kernel-worthy asymptotic partial survived.

**Scope.**

The now-proved positive-power lower question is excluded. Prime gaps use zero-indexed nth primes; runs are pairwise distinct as sets of indices; the record maximizes over starts below x.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Nat.Prime.Nth
import Mathlib.Order.Filter.AtTopBot.Basic

open Filter
open scoped Topology

/-!
# Erdős problem 852, remaining open upper-bound question

Is the longest run of pairwise distinct consecutive prime gaps among starting
indices below `x` of length `o(log x)`?
-/

namespace Statements.Erdos852DistinctPrimeGaps

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

def DistinctRun (start length : ℕ) : Prop :=
  (Finset.range length : Set ℕ).Pairwise fun i j =>
    primeGap (start + i) ≠ primeGap (start + j)

noncomputable def longestRun (x : ℕ) : ℕ :=
  sSup {length : ℕ | ∃ start < x, DistinctRun start length}

abbrev statement : Prop :=
  Tendsto
    (fun x : ℕ => (longestRun x : ℝ) / Real.log x)
    atTop (𝓝 0)

theorem target : statement := sorry

end Statements.Erdos852DistinctPrimeGaps
```

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