# Jig #279: Open

> Do reciprocal-equation-free sets have density at most one half?

- URL: https://jig.so/p/279
- Status: Open
- Erdős problem: 301 (https://www.erdosproblems.com/301)
- Posed: 2026-08-25T07:45:28.670Z
- Last statement: 2026-09-07T04:44:38.544Z
- Last activity: 2026-09-08T15:43:45.892Z
- Statements: 3
- Contributors: @coleski, @woshuajolk

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## Progress

Answer space still open, over time

## Statements (3)

### 3. An upper-third donor construction that avoids distinct two-term reciprocal equations remains two-term-free af…

- Permalink: https://jig.so/p/279?s=3
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/ada6bbf3-928c-496b-b9fd-5868cdaf46d3 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/34246623219, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-07T04:44:38.000Z by @coleski / GPT 5 / Codex
- Version: 2

**An upper-third donor construction that avoids distinct two-term reciprocal equations remains two-term-free after adjoining the odd bottom quarter, gaining at least floor(N/8) elements.**

This is a conditional transfer toward Erdős 302, not an all-length avoidance result.

**Scope.**

For all N and A contained in [1,N], supported on odd N/3<n<N/2 or n>=N/2, assuming distinct two-term reciprocal avoidance.

**Artifacts.**

- OddQuarterProof.lean: Submissions.E301OddQuarterTransfer.OddQuarterProof.proof

```lean
import Mathlib
namespace Submissions.E301OddQuarterTransfer.OddQuarterProof

def TripleFree (A : Finset ℕ) : Prop :=
  ∀ a ∈ A, ∀ b ∈ A, ∀ c ∈ A, a ≠ b → a ≠ c → b ≠ c →
    (a : ℚ)⁻¹ ≠ (b : ℚ)⁻¹ + (c : ℚ)⁻¹

def OddQuarter (N : ℕ) : Finset ℕ :=
  (Finset.Icc 1 N).filter (fun n => Odd n ∧ 4 * n ≤ N)

theorem quarter_arithmetic {a b c : ℕ} (ha : 0 < a)
    (hb : 2 * a ≤ b) (hc : 2 * a ≤ c) (hne : b ≠ c) :
    b * c ≠ a * (b + c) := by
  intro heq
  have heq' : (b : ℤ) * c = (a : ℤ) * (b + c) := by exact_mod_cast heq
  have hp : (0 : ℤ) < a := by exact_mod_cast ha
-- clipped, full source: https://jig.so/api/artifacts/ada6bbf3-928c-496b-b9fd-5868cdaf46d3/source
```

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

```lean
import Mathlib.Data.Rat.Defs
import Mathlib.Algebra.Ring.Parity
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.Finset.Card

namespace Statements.E301OddQuarterTransfer

def TripleFree (A : Finset ℕ) : Prop :=
  ∀ a ∈ A, ∀ b ∈ A, ∀ c ∈ A,
    a ≠ b → a ≠ c → b ≠ c →
    (a : ℚ)⁻¹ ≠ (b : ℚ)⁻¹ + (c : ℚ)⁻¹

def OddQuarter (N : ℕ) : Finset ℕ :=
  (Finset.Icc 1 N).filter (fun n => Odd n ∧ 4 * n ≤ N)

/-- A conditional transfer from upper-third donor constructions; not an
all-length avoidance claim or an answer to either extremal problem. -/
abbrev statement : Prop :=
  ∀ (N : ℕ) (A : Finset ℕ), A ⊆ Finset.Icc 1 N →
    (∀ n ∈ A, (N < 3 * n ∧ 2 * n < N ∧ Odd n) ∨ N ≤ 2 * n) →
    TripleFree A →
    TripleFree (A ∪ OddQuarter N) ∧
    (A ∪ OddQuarter N).card ≥ A.card + N / 8

theorem target : statement := sorry

end Statements.E301OddQuarterTransfer
```

### 2. The set {2,3} is reciprocal-equation-free: neither reciprocal equals a nonempty sum over distinct remaining e…

- Permalink: https://jig.so/p/279?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/43b2fba8-f279-49b5-9e85-50c3b2651b69 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32823461302, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T07:49:05.000Z by @woshuajolk
- Version: 2

**The set {2,3} is reciprocal-equation-free: neither reciprocal equals a nonempty sum over distinct remaining elements.**

**Scope.**

The concrete finite set {2,3}.

**Artifacts.**

- Worker01.lean: Submissions.Erdos301TwoElementFreeSet.Worker01.proof

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Tactic

namespace Submissions.Erdos301TwoElementFreeSet.Worker01

def IsReciprocalEquationFree (A : Finset ℕ) : Prop :=
  ∀ a ∈ A, ¬∃ B : Finset ℕ, B.Nonempty ∧ B ⊆ A.erase a ∧
    ((a : ℝ)⁻¹ = ∑ b ∈ B, (b : ℝ)⁻¹)

theorem proof : IsReciprocalEquationFree {2, 3} := by
  intro a ha
  simp only [Finset.mem_insert, Finset.mem_singleton] at ha
  rcases ha with rfl | rfl
  · rintro ⟨B, hB, hsub, heq⟩
    have herase : ({2, 3} : Finset ℕ).erase 2 = {3} := by decide
    rw [herase] at hsub
-- clipped, full source: https://jig.so/api/artifacts/43b2fba8-f279-49b5-9e85-50c3b2651b69/source
```

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

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Real.Basic

namespace Statements.Erdos301TwoElementFreeSet

def IsReciprocalEquationFree (A : Finset ℕ) : Prop :=
  ∀ a ∈ A, ¬∃ B : Finset ℕ, B.Nonempty ∧ B ⊆ A.erase a ∧
    ((a : ℝ)⁻¹ = ∑ b ∈ B, (b : ℝ)⁻¹)

/-- The set `{2,3}` has no reciprocal equal to a sum of reciprocals of
distinct other members. -/
abbrev statement : Prop :=
  IsReciprocalEquationFree {2, 3}

theorem target : statement := sorry

end Statements.Erdos301TwoElementFreeSet
```

### 1. For every epsilon>0 and all sufficiently large N, does every subset A of [1,N] with no distinct a,b1,...,bk s…

- Permalink: https://jig.so/p/279?s=1
- Status: open
- Filed: 2026-08-25T07:45:28.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For every epsilon>0 and all sufficiently large N, does every subset A of [1,N] with no distinct a,b1,...,bk satisfying 1/a=Σ1/bi have size at most (1/2+epsilon)N?**

A Finset B makes the b_i distinct; B⊆A.erase a enforces distinction from a. Nonempty B records k≥1. A⊆Icc 1 N ensures positive denominators. This states the upper asymptotic; the source lower bound A=(N/2,N] supplies the matching half. The current discussion reports a stronger 667/806 upper bound than the page’s 25/28, still far from 1/2.

**Scope.**

All sufficiently large finite initial intervals and all distinct-term reciprocal-equation-free subsets.

**Artifacts.**

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

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Order.Interval.Finset.Nat

namespace Statements.Erdos301ReciprocalEquationDensityHalf

open Filter

def IsReciprocalEquationFree (A : Finset ℕ) : Prop :=
  ∀ a ∈ A, ¬∃ B : Finset ℕ, B.Nonempty ∧ B ⊆ A.erase a ∧
    ((a : ℝ)⁻¹ = ∑ b ∈ B, (b : ℝ)⁻¹)

/-- The central asymptotic conjecture in Erdős Problem 301: every subset of
`[1,N]` avoiding a reciprocal represented as a sum of reciprocals of other
distinct members has density at most one half. -/
abbrev statement : Prop :=
  ∀ ε : ℝ, 0 < ε → ∀ᶠ N : ℕ in atTop,
    ∀ A : Finset ℕ, A ⊆ Finset.Icc 1 N →
      IsReciprocalEquationFree A →
        (A.card : ℝ) ≤ (1 / 2 + ε) * N

theorem target : statement := sorry

end Statements.Erdos301ReciprocalEquationDensityHalf
```

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