# 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
- 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
  have hb' : (0 : ℤ) ≤ (b : ℤ) - 2 * a := by omega
  have hc' : (0 : ℤ) ≤ (c : ℤ) - 2 * a := by omega
  have hs : (0 : ℤ) < (b : ℤ) + c - 4 * a := by omega
  nlinarith [mul_nonneg hb' hc', mul_pos hp hs]

theorem proof : ∀ (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 := by
  classical
  intro N A hA hshape hfree
  have hoq : ∀ n ∈ OddQuarter N, 0 < n ∧ Odd n ∧ 4 * n ≤ N := by
    intro n hn
    have h := Finset.mem_filter.mp hn
    exact ⟨(Finset.mem_Icc.mp h.1).1, h.2⟩
  have hpos : ∀ n ∈ A ∪ OddQuarter N, 0 < n := by
    intro n hn
    rcases Finset.mem_union.mp hn with hn | hn
    · exact (Finset.mem_Icc.mp (hA hn)).1
    · exact (hoq n hn).1
  constructor
  · intro a ha b hb c hc hab hac hbc heq
    have pa := hpos a ha
    have pb := hpos b hb
    have pc := hpos c hc
    have za : (a : ℚ) ≠ 0 := by positivity
    have zb : (b : ℚ) ≠ 0 := by positivity
    have zc : (c : ℚ) ≠ 0 := by positivity
    have eqQ : (b : ℚ) * c = a * (b + c) := by
      field_simp at heq
      nlinarith [heq]
    have eqN : b*c = a*(b+c) := by exact_mod_cast eqQ
    have ab : a < b := by
      by_contra h
      have : b ≤ a := by omega
      nlinarith
    have ac : a < c := by
      by_contra h
      have : c ≤ a := by omega
      nlinarith
    have small : 4*a ≤ N := by
      by_contra h
      have allA : ∀ x ∈ A ∪ OddQuarter N, a ≤ x → x ∈ A := by
        intro x hx hax
        rcases Finset.mem_union.mp hx with hx | hx
        · exact hx
        · have := (hoq x hx).2.2; omega
      exact hfree a (allA a ha (by omega)) b (allA b hb (by omega))
        c (allA c hc (by omega)) hab hac hbc heq
    have oa : Odd a := by
      rcases Finset.mem_union.mp ha with ha | ha
      · rcases hshape a ha with h | h
        · exact h.2.2
        · omega
      · exact (hoq a ha).2.1
    have amod : a % 2 = 1 := Nat.odd_iff.mp oa
    have eqmod := congrArg (fun x : ℕ => x % 2) eqN
    rw [Nat.mul_mod b c, Nat.mul_mod a (b+c), Nat.add_mod b c, amod] at eqmod
    have mods : b % 2 = 0 ∧ c % 2 = 0 := by
      have bh := Nat.mod_lt b (by decide : 0 < 2)
      have ch := Nat.mod_lt c (by decide : 0 < 2)
      have hbmod : b % 2 = 0 ∨ b % 2 = 1 := by omega
      have hcmod : c % 2 = 0 ∨ c % 2 = 1 := by omega
      rcases hbmod with hbmod | hbmod <;> rcases hcmod with hcmod | hcmod <;>
        simp_all
    obtain ⟨bm, cm⟩ := mods
    have large : ∀ x ∈ A ∪ OddQuarter N, x % 2 = 0 → N ≤ 2*x := by
      intro x hx hm
      rcases Finset.mem_union.mp hx with hx | hx
      · rcases hshape x hx with h | h
        · have := Nat.odd_iff.mp h.2.2; omega
        · exact h
      · have := Nat.odd_iff.mp (hoq x hx).2.1; omega
    exact quarter_arithmetic pa (by have := large b hb bm; omega)
      (by have := large c hc cm; omega) hbc eqN
  · have hd : Disjoint A (OddQuarter N) := by
      apply Finset.disjoint_left.mpr
      intro n hn ho
      have hp := hoq n ho
      rcases hshape n hn with h | h <;> omega
    have hsub : (Finset.range (N/8)).image (fun k => 2*k+1) ⊆ OddQuarter N := by
      intro n hn
      obtain ⟨k,hk,rfl⟩ := Finset.mem_image.mp hn
      have := Finset.mem_range.mp hk
      apply Finset.mem_filter.mpr
      exact ⟨Finset.mem_Icc.mpr ⟨by omega, by omega⟩, ⟨k, by omega⟩, by omega⟩
    have hcard : ((Finset.range (N/8)).image (fun k => 2*k+1)).card = N/8 := by
      rw [Finset.card_image_of_injective]
      · exact Finset.card_range _
      · intro a b h
        change 2*a+1 = 2*b+1 at h
        omega
    have := Finset.card_le_card hsub
    rw [hcard] at this
    rw [Finset.card_union_of_disjoint hd]
    omega
end Submissions.E301OddQuarterTransfer.OddQuarterProof
```

- Canonical statement

```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
- 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
    obtain ⟨x, hx⟩ := hB
    have hx3 : x = 3 := by
      have := hsub hx
      simpa using this
    subst x
    have hB3 : B = {3} :=
      Finset.Subset.antisymm hsub (Finset.singleton_subset_iff.mpr hx)
    subst B
    norm_num at heq
  · rintro ⟨B, hB, hsub, heq⟩
    have herase : ({2, 3} : Finset ℕ).erase 3 = {2} := by decide
    rw [herase] at hsub
    obtain ⟨x, hx⟩ := hB
    have hx2 : x = 2 := by
      have := hsub hx
      simpa using this
    subst x
    have hB2 : B = {2} :=
      Finset.Subset.antisymm hsub (Finset.singleton_subset_iff.mpr hx)
    subst B
    norm_num at heq

end Submissions.Erdos301TwoElementFreeSet.Worker01
```

- Canonical statement

```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

```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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