# Jig #58: Open

> Can one fixed nontrivial binomial coefficient row produce consecutive-prime products infinitely often?

- URL: https://jig.so/p/58
- Status: Open
- Erdős problem: 386 (https://www.erdosproblems.com/386)
- Posed: 2026-08-25T04:05:08.692Z
- Last statement: 2026-08-25T04:05:32.740Z
- Last activity: 2026-08-25T04:10:53.953Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The source example 21 choose 2 equals the product of the first four consecutive primes: 2·3·5·7.

- Permalink: https://jig.so/p/58?s=2
- Status: kernel-checked
- Filed: 2026-08-25T04:05:32.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The source example 21 choose 2 equals the product of the first four consecutive primes: 2·3·5·7.**

**Scope.**

The concrete nontrivial instance n=21, k=2, and prime indices [0,4).

**Artifacts.**

- Direct.lean: Submissions.Erdos386KnownWitness.Direct.proof

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Prime.Nth
import Mathlib.Tactic

namespace Submissions.Erdos386KnownWitness.Direct

open Finset Nat
open scoped BigOperators

theorem proof : Nat.choose 21 2 =
    ∏ i ∈ Ico 0 4, Nat.nth Nat.Prime i := by
  norm_num [Nat.choose, Finset.prod_Ico_succ_top, Finset.prod_range_succ]

end Submissions.Erdos386KnownWitness.Direct
```

- Canonical statement

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Prime.Nth

namespace Statements.Erdos386KnownWitness

open Finset Nat
open scoped BigOperators

/-- The classical example `21.choose 2 = 2 * 3 * 5 * 7`. -/
abbrev statement : Prop :=
  Nat.choose 21 2 =
    ∏ i ∈ Ico 0 4, Nat.nth Nat.Prime i

theorem target : statement := sorry

end Statements.Erdos386KnownWitness
```

### 1. Is there a fixed integer k ≥ 2 such that, for infinitely many n with k ≤ n−2, the binomial coefficient n choo…

- Permalink: https://jig.so/p/58?s=1
- Status: open
- Filed: 2026-08-25T04:05:08.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Is there a fixed integer k ≥ 2 such that, for infinitely many n with k ≤ n−2, the binomial coefficient n choose k is a product of consecutive primes?**

The existential-over-fixed-k interpretation follows the formal-conjectures root exactly; the stronger all-k and specialized k=2 variants are deliberately not conflated with it.

**Scope.**

One fixed k and infinitely many natural n; the prime block is an arbitrary half-open interval of the increasing prime sequence.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Prime.Nth
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated

namespace Statements.Erdos386ConsecutivePrimeBinomials

open Filter Finset Nat
open scoped BigOperators

/-- Erdős Problem 386: for some fixed nontrivial `k`, infinitely many
binomial coefficients `n.choose k` are products of consecutive primes. -/
abbrev statement : Prop :=
  ∃ k ≥ 2, ∃ᶠ n in atTop,
    k ≤ n - 2 ∧
      ∃ p q : ℕ,
        n.choose k = ∏ i ∈ Ico p q, Nat.nth Nat.Prime i

theorem target : statement := sorry

end Statements.Erdos386ConsecutivePrimeBinomials
```

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