# Jig #180: Prior art

> Are infinitely many central-binomial pairs equal in prime support?

- URL: https://jig.so/p/180
- Status: Prior art
- Erdős problem: 730 (https://www.erdosproblems.com/730)
- Prior art: https://github.com/williamjblair/lean-proofs/blob/5d10b4d91f257cfbe8c563cf927f543a868845e0/ErdosProblems/Erdos730/FullDensityTheorem.lean — Existing full theorem Erdos730.FullDensityTheorem.pairSet_infinite, Lean formalization by William Blair of the argument credited to Liam Price (June24,2026). This port amalgamates its full transitive proof cone for exact Jig verification.
- Posed: 2026-08-25T06:27:35.315Z
- Last statement: 2026-08-25T06:27:50.359Z
- Last activity: 2026-09-05T22:26:32.831Z
- Statements: 2
- Contributors: @woshuajolk, @declangessel

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## Progress

Answer space still open, over time

## Resolution

- Solved: elsewhere, and verified here
- Closed for: All natural indices n<m and exact equality of finite prime-divisor sets.
- By: @declangessel, @woshuajolk

- BlairPort.lean: Submissions.Erdos730CentralBinomPrimeSupport.BlairPort.proof — axioms clean

```lean
/-
Full proof port of William Blair's public Erdős 730 formalization.
Original informal argument: Liam Price, June 24, 2026.
Analytic dependency: PrimeNumberTheoremAnd and its contributors.
Original source: https://github.com/williamjblair/lean-proofs
Pinned Lean 4.33.0 and mathlib db584cd6d46c92f209a44c0f1c829460d327499d.
PNT dependency: ajirving/PrimeNumberTheoremAnd at commit 769d3b81fb.
This is a prior-art proof port, not a claim of a new mathematical solution.
Original author comments and mathematical proof bodies are retained below.
Blueprint metadata, two unused admitted PNT declarations and the unused dependent decay_alt are omit
-/
/-
Upstream license: Will Blair (MIT)
MIT License

Copyright (c) 2026 Will Blair
-- clipped, full source: https://jig.so/api/artifacts/0f2a4e7d-73c6-4b70-8053-f0cea2a3b4cc/source
```

### Verification

Read from the verifier's report on the closing proof.

- Verdict report: [artifact 0f2a4e7d](https://jig.so/api/artifacts/0f2a4e7d-73c6-4b70-8053-f0cea2a3b4cc)
- CI run: [GitHub Actions log](https://github.com/WoshuaJolk/jig-verifier/actions/runs/33995921974)
- Checks passed: build, axioms, manifest, no new axioms, static policy, anti restatement
- Axioms: Classical.choice, Quot.sound, propext
- Toolchain: Lean v4.33.0, Mathlib db584cd (Jig's pins)

## Statements (2)

### 2. The index pairs (87,88) and (607,608) each give central binomial coefficients with exactly the same prime div…

- Permalink: https://jig.so/p/180?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/bcf99fe5-a53e-4847-928b-023ea72a0301 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32817302282, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T06:27:50.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**The index pairs (87,88) and (607,608) each give central binomial coefficients with exactly the same prime divisors.**

**Scope.**

Two exact ordered pairs, checked against the same primeFactors predicate as the root.

**Artifacts.**

- KernelCheck.lean: Submissions.Erdos730ExplicitPrimeSupportPairs.KernelCheck.proof

```lean
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic

namespace Submissions.Erdos730ExplicitPrimeSupportPairs.KernelCheck

abbrev S : Set (ℕ × ℕ) :=
  {(n, m) : ℕ × ℕ | n < m ∧
    n.centralBinom.primeFactors = m.centralBinom.primeFactors}

theorem proof :
    ({(87, 88), (607, 608)} : Set (ℕ × ℕ)) ⊆ S := by
  have h87a : Nat.centralBinom 87 ∣ Nat.centralBinom 88 ^ 6 := by
    norm_num [Nat.centralBinom, Nat.choose_eq_descFactorial_div_factorial]
  have h87b : Nat.centralBinom 88 ∣ Nat.centralBinom 87 ^ 6 := by
    norm_num [Nat.centralBinom, Nat.choose_eq_descFactorial_div_factorial]
-- clipped, full source: https://jig.so/api/artifacts/bcf99fe5-a53e-4847-928b-023ea72a0301/source
```

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

```lean
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic

namespace Statements.Erdos730ExplicitPrimeSupportPairs

abbrev S : Set (ℕ × ℕ) :=
  {(n, m) : ℕ × ℕ | n < m ∧
    n.centralBinom.primeFactors = m.centralBinom.primeFactors}

/-- Two exact pairs of indices whose central binomial coefficients have the same prime support. -/
abbrev statement : Prop :=
  ({(87, 88), (607, 608)} : Set (ℕ × ℕ)) ⊆ S

theorem target : statement := sorry

end Statements.Erdos730ExplicitPrimeSupportPairs
```

### 1. There are infinitely many ordered pairs n<m for which the central binomial coefficients C(2n,n) and C(2m,m) h…

- Permalink: https://jig.so/p/180?s=1
- Status: prior art
- Verified: green proof https://jig.so/api/artifacts/0f2a4e7d-73c6-4b70-8053-f0cea2a3b4cc (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/33995921974, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T06:27:35.000Z by @woshuajolk, @declangessel / GPT 5.6 Sol / Cursor
- Version: 3
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**There are infinitely many ordered pairs n<m for which the central binomial coefficients C(2n,n) and C(2m,m) have exactly the same prime divisors.**

**Scope.**

All natural indices n<m and exact equality of finite prime-divisor sets.

**Artifacts.**

- BlairPort.lean: Submissions.Erdos730CentralBinomPrimeSupport.BlairPort.proof

```lean
/-
Full proof port of William Blair's public Erdős 730 formalization.
Original informal argument: Liam Price, June 24, 2026.
Analytic dependency: PrimeNumberTheoremAnd and its contributors.
Original source: https://github.com/williamjblair/lean-proofs
Pinned Lean 4.33.0 and mathlib db584cd6d46c92f209a44c0f1c829460d327499d.
PNT dependency: ajirving/PrimeNumberTheoremAnd at commit 769d3b81fb.
This is a prior-art proof port, not a claim of a new mathematical solution.
Original author comments and mathematical proof bodies are retained below.
Blueprint metadata, two unused admitted PNT declarations and the unused dependent decay_alt are omit
-/
/-
Upstream license: Will Blair (MIT)
MIT License

Copyright (c) 2026 Will Blair
-- clipped, full source: https://jig.so/api/artifacts/0f2a4e7d-73c6-4b70-8053-f0cea2a3b4cc/source
```

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

```lean
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic

namespace Statements.Erdos730CentralBinomPrimeSupport

abbrev S : Set (ℕ × ℕ) :=
  {(n, m) : ℕ × ℕ | n < m ∧
    n.centralBinom.primeFactors = m.centralBinom.primeFactors}

/-- Erdős Problem 730: infinitely many distinct central binomial coefficients have identical prime support in pairs. -/
abbrev statement : Prop :=
  S.Infinite

theorem target : statement := sorry

end Statements.Erdos730CentralBinomPrimeSupport
```

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