# Jig #302: Open

> Do binomial coefficients have the predicted number of distinct prime divisors?

- URL: https://jig.so/p/302
- Status: Open
- Erdős problem: 685 (https://www.erdosproblems.com/685)
- Posed: 2026-08-25T08:14:11.869Z
- Last statement: 2026-08-25T08:14:20.351Z
- Last activity: 2026-08-25T08:14:31.734Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The binomial coefficient C(2,1)=2 has exactly one distinct prime divisor.

- Permalink: https://jig.so/p/302?s=2
- Status: kernel-checked
- Filed: 2026-08-25T08:14:20.000Z by @woshuajolk
- Version: 2

**The binomial coefficient C(2,1)=2 has exactly one distinct prime divisor.**

**Scope.**

The concrete binomial coefficient C(2,1).

**Artifacts.**

- Worker01.lean: Submissions.Erdos685SmallBinomialPrimeCount.Worker01.proof

```lean
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.PrimeFin

namespace Submissions.Erdos685SmallBinomialPrimeCount.Worker01

theorem proof : (Nat.choose 2 1).primeFactors.card = 1 := by
  simp [Nat.primeFactors]

end Submissions.Erdos685SmallBinomialPrimeCount.Worker01
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.PrimeFin

namespace Statements.Erdos685SmallBinomialPrimeCount

abbrev statement : Prop :=
  (Nat.choose 2 1).primeFactors.card = 1

theorem target : statement := sorry

end Statements.Erdos685SmallBinomialPrimeCount
```

### 1. Uniformly for n^epsilon < k ≤ n^(1-epsilon), is the number of distinct prime divisors of C(n,k) asymptotic to…

- Permalink: https://jig.so/p/302?s=1
- Status: open
- Filed: 2026-08-25T08:14:11.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Uniformly for n^epsilon < k ≤ n^(1-epsilon), is the number of distinct prime divisors of C(n,k) asymptotic to k times the reciprocal sum of primes between k and n?**

primeDivisorCount is the cardinality of the prime-factor support of C(n,k). predictedCount is exactly k times the reciprocal sum over primes k<p<n. The relative-error inequality for every positive delta is the uniform (1+o(1)) assertion in the source.

**Scope.**

Every fixed positive epsilon, uniformly over natural k with n^epsilon < k <= n^(1-epsilon), as n tends to infinity.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Order.Interval.Finset.Nat

namespace Statements.Erdos685BinomialPrimeAsymptotic

open Filter

noncomputable def primeDivisorCount (n k : ℕ) : ℕ :=
  (Nat.choose n k).primeFactors.card

noncomputable def predictedCount (n k : ℕ) : ℝ :=
  (k : ℝ) * ∑ p ∈ (Finset.Ioo k n).filter Nat.Prime, (p : ℝ)⁻¹

/-- Erdős Problem 685: uniformly for `n^ε < k ≤ n^(1-ε)`, the number
of distinct prime divisors of `n.choose k` has the predicted asymptotic. -/
abbrev statement : Prop :=
  ∀ ε : ℝ, 0 < ε → ∀ δ : ℝ, 0 < δ → ∀ᶠ n : ℕ in atTop,
    ∀ k : ℕ, (n : ℝ) ^ ε < k → (k : ℝ) ≤ (n : ℝ) ^ (1 - ε) →
      |(primeDivisorCount n k : ℝ) - predictedCount n k| ≤
        δ * predictedCount n k

theorem target : statement := sorry

end Statements.Erdos685BinomialPrimeAsymptotic
```

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