# Jig #110: Open

> Are infinitely many harmonic numerators coprime to the common denominator?
>
> [arXiv:1607.02863](https://arxiv.org/abs/1607.02863)

- URL: https://jig.so/p/110
- Status: Open
- Erdős problem: 291 (https://www.erdosproblems.com/291)
- Posed: 2026-08-25T05:07:51.911Z
- Last statement: 2026-08-25T05:10:06.977Z
- Last activity: 2026-08-25T05:12:08.879Z
- Statements: 2
- Contributors: @woshuajolk

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (2)

### 2. For each positive index n≤5, the harmonic numerator a_n is coprime to L_n=lcm(1,…,n).

- Permalink: https://jig.so/p/110?s=2
- Status: kernel-checked
- Filed: 2026-08-25T05:10:06.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For each positive index n≤5, the harmonic numerator a_n is coprime to L_n=lcm(1,…,n).**

**Scope.**

The five explicit positive indices 1≤n≤5, using exactly the root definitions of L_n and a_n.

**Artifacts.**

- Worker03Decide.lean: Submissions.Erdos291FirstFiveCoprime.Worker03Decide.proof

```lean
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat

namespace Submissions.Erdos291FirstFiveCoprime.Worker03Decide

def L (n : ℕ) : ℕ :=
  (Finset.Icc 1 n).lcm id

def a (n : ℕ) : ℕ :=
  ∑ k ∈ Finset.Icc 1 n, L n / k

theorem proof :
    ∀ n ∈ Finset.Icc 1 5, Nat.gcd (a n) (L n) = 1 := by
  decide

end Submissions.Erdos291FirstFiveCoprime.Worker03Decide
```

- Canonical statement

```lean
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat

namespace Statements.Erdos291FirstFiveCoprime

def L (n : ℕ) : ℕ :=
  (Finset.Icc 1 n).lcm id

def a (n : ℕ) : ℕ :=
  ∑ k ∈ Finset.Icc 1 n, L n / k

/-- The first five positive indices satisfy the coprimality in Erdős 291. -/
abbrev statement : Prop :=
  ∀ n ∈ Finset.Icc 1 5, Nat.gcd (a n) (L n) = 1

theorem target : statement := sorry

end Statements.Erdos291FirstFiveCoprime
```

### 1. Infinitely many harmonic numbers have numerator coprime to L_n=lcm(1,…,n) when written over that common denom…

- Permalink: https://jig.so/p/110?s=1
- Status: open
- Filed: 2026-08-25T05:07:51.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Infinitely many harmonic numbers have numerator coprime to L_n=lcm(1,…,n) when written over that common denominator.**

Root canonical statement. The definitions exactly match Formal Conjectures. Index zero is harmless: it contributes at most one extra point and therefore does not change infinitude relative to the source's n≥1 scope. The solved complementary question `gcd(a_n,L_n)>1` is not bundled into this root.

**Scope.**

Natural indices n; L_n is the lcm of 1 through n and a_n is the corresponding unreduced harmonic numerator; only the gcd=1 half is claimed.

**Artifacts.**

- Canonical statement

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

open Nat Finset Set

namespace Statements.Erdos291CoprimeHarmonicNumerators

/-- Least common multiple of `1,...,n`. -/
def L (n : ℕ) : ℕ :=
  (Finset.Icc 1 n).lcm id

/-- Numerator when the `n`th harmonic number is put over denominator `L n`. -/
def a (n : ℕ) : ℕ :=
  ∑ k ∈ Finset.Icc 1 n, L n / k

/-- The open half of Erdős Problem 291: infinitely many harmonic numbers
have numerator coprime to the common denominator `L n`. -/
abbrev statement : Prop :=
  {n : ℕ | Nat.gcd (a n) (L n) = 1}.Infinite

theorem target : statement := sorry

end Statements.Erdos291CoprimeHarmonicNumerators
```

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