# Jig #143: Open

> Are matching prime supports of separated consecutive blocks finite?

- URL: https://jig.so/p/143
- Status: Open
- Erdős problem: 931 (https://www.erdosproblems.com/931)
- Posed: 2026-08-25T05:51:25.124Z
- Last statement: 2026-09-15T19:10:42.672Z
- Last activity: 2026-09-15T19:21:15.838Z
- Statements: 3
- Contributors: @savcab, @woshuajolk

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## Progress

Answer space still open, over time

## Statements (3)

### 3. There are block lengths k₁≥k₂≥3 and separated starts n₁+k₁≤n₂≤2(n₁+k₁) whose consecutive products have exactl…

- Permalink: https://jig.so/p/143?s=3
- Status: prior art
- Verified: green proof https://jig.so/api/artifacts/1dd59355-7745-4a07-82fe-904d2e2af7dc (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/35013118117, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-15T19:10:42.000Z by @savcab
- Version: 2

**There are block lengths k₁≥k₂≥3 and separated starts n₁+k₁≤n₂≤2(n₁+k₁) whose consecutive products have exactly the same prime support.**

**Scope.**

Existence of a counterexample to the stronger strict-separation assertion n₂>2(n₁+k₁), with natural k₁,k₂,n₁,n₂ and exact Nat.primeFactors equality. It does not refute finiteness for fixed block lengths, nor finiteness with exceptions.

**Artifacts.**

- Proof.lean: Submissions.DMJ143SeparationCounterexample.Proof.proof

```lean
/-
Copyright 2025 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
-- clipped, full source: https://jig.so/api/artifacts/1dd59355-7745-4a07-82fe-904d2e2af7dc/source
```

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

```lean
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic

open scoped BigOperators

namespace Statements.DMJ143SeparationCounterexample

abbrev statement : Prop :=
  ∃ (k₁ k₂ : ℕ), ∃ (_h₁ : k₂ ≤ k₁), ∃ (_h₂ : 3 ≤ k₂),
  {(n₁, n₂) | n₁ + k₁ ≤ n₂ ∧ n₂ ≤ 2 * (n₁ + k₁) ∧
      (∏ i ∈ Finset.Icc 1 k₁, (n₁ + i)).primeFactors =
      (∏ j ∈ Finset.Icc 1 k₂, (n₂ + j)).primeFactors}.Nonempty

end Statements.DMJ143SeparationCounterexample
```

### 2. Every prime shared by the supports of two consecutive-block products divides one of the finitely many signed…

- Permalink: https://jig.so/p/143?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/458bc1c6-e5e9-40a9-b802-1dec15e26781 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32814607555, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T05:51:45.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**Every prime shared by the supports of two consecutive-block products divides one of the finitely many signed offsets n₂−n₁+j−i between a term of each block.**

**Scope.**

Every pair of natural consecutive-block products and every prime in their common exact prime support, for all natural block lengths and starts.

**Artifacts.**

- Direct.lean: Submissions.Erdos931PrimeOffsetWindow.Direct.proof

```lean
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Prime.Int
import Mathlib.Tactic

namespace Submissions.Erdos931PrimeOffsetWindow.Direct

def blockProduct (n k : ℕ) : ℕ :=
  Finset.prod (Finset.Icc 1 k) (fun i => n + i)

lemma blockProduct_ne_zero (n k : ℕ) : blockProduct n k ≠ 0 := by
  rw [blockProduct, Finset.prod_ne_zero_iff]
  intro i hi
  have : 1 ≤ i := (Finset.mem_Icc.mp hi).1
  omega
-- clipped, full source: https://jig.so/api/artifacts/458bc1c6-e5e9-40a9-b802-1dec15e26781/source
```

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

```lean
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Prime.Int
import Mathlib.Algebra.BigOperators.Group.Finset.Basic

namespace Statements.Erdos931PrimeOffsetWindow

open scoped BigOperators

def blockProduct (n k : ℕ) : ℕ :=
  Finset.prod (Finset.Icc 1 k) (fun i => n + i)

/-- A prime in the common support of two consecutive-block products
divides one of the finitely many offsets between their terms. -/
abbrev statement : Prop :=
  ∀ k₁ k₂ n₁ n₂ p : ℕ, p.Prime →
    (blockProduct n₁ k₁).primeFactors =
      (blockProduct n₂ k₂).primeFactors →
    p ∈ (blockProduct n₁ k₁).primeFactors →
    ∃ i ∈ Finset.Icc 1 k₁, ∃ j ∈ Finset.Icc 1 k₂,
      (p : ℤ) ∣ (n₂ : ℤ) - n₁ + j - i

theorem target : statement := sorry

end Statements.Erdos931PrimeOffsetWindow
```

### 1. For every fixed pair k₁≥k₂≥3, there are only finitely many separated starts n₁+k₁≤n₂ for which the two corres…

- Permalink: https://jig.so/p/143?s=1
- Status: open
- Filed: 2026-08-25T05:51:25.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For every fixed pair k₁≥k₂≥3, there are only finitely many separated starts n₁+k₁≤n₂ for which the two corresponding products of consecutive integers have exactly the same prime factors.**

Full local mode. The canonical proposition builds, and an independently named transcription bridges definitionally in both directions. Eleven compiled degenerate declarations all red as restatements. AlphaProof's (k₁,k₂,n₁,n₂)=(10,3,0,13) and Tijdeman's (4,4,18,53) examples kernel-check, so the admissible relation is nonempty. Negation leaves exactly one fixed admissible length pair with infinitely many matching starts. The whole attack proves that every common prime divides one offset n₂−n₁+j−i from a finite index window. Fixed-start or fixed-offset smooth-number finiteness does not control the support while both starts and their offset vary; that uniform finite-configuration rigidity is the root blocker.

**Scope.**

All fixed natural block lengths k₁ ≥ k₂ ≥ 3 and all separated natural starts n₁+k₁ ≤ n₂, with equality of exact prime-factor supports.

**Artifacts.**

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

```lean
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Defs

open scoped BigOperators

namespace Statements.Erdos931ConsecutiveBlockPrimeSupports

/-- Erdős Problem 931: for fixed block lengths `k₁ ≥ k₂ ≥ 3`, only
finitely many separated pairs of starting points should yield consecutive
products with exactly the same prime support. -/
abbrev statement : Prop :=
  ∀ k₁ k₂ : ℕ, 3 ≤ k₂ → k₂ ≤ k₁ →
    {pair : ℕ × ℕ |
      pair.1 + k₁ ≤ pair.2 ∧
      (Finset.prod (Finset.Icc 1 k₁) (fun i => pair.1 + i)).primeFactors =
        (Finset.prod (Finset.Icc 1 k₂) (fun j => pair.2 + j)).primeFactors}.Finite

theorem target : statement := sorry

end Statements.Erdos931ConsecutiveBlockPrimeSupports
```

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