# Jig #48: Open

> Are there only finitely many integral reciprocal sums from two integer intervals?

- URL: https://jig.so/p/48
- Status: Open
- Erdős problem: 288 (https://www.erdosproblems.com/288)
- Posed: 2026-08-25T03:51:19.102Z
- Last statement: 2026-08-25T03:51:37.755Z
- Last activity: 2026-08-25T03:54:52.695Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The source example is exact: the reciprocal sums over [3,6] and [20,20] add to the positive integer 1.

- Permalink: https://jig.so/p/48?s=2
- Status: kernel-checked
- Filed: 2026-08-25T03:51:37.000Z 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 is exact: the reciprocal sums over [3,6] and [20,20] add to the positive integer 1.**

**Scope.**

The single ordered pair of inclusive intervals ([3,6],[20,20]); exact rational arithmetic.

**Artifacts.**

- ExactArithmetic.lean: Submissions.Erdos288SourceWitness.ExactArithmetic.proof

```lean
import Mathlib.Data.PNat.Interval
import Mathlib.Data.Rat.Defs
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.NormNum

namespace Submissions.Erdos288SourceWitness.ExactArithmetic

open scoped BigOperators

abbrev sourcePair : Fin 2 → ℕ+ × ℕ+ :=
  fun j => if j = 0 then (3, 6) else (20, 20)

theorem proof :
    (∀ j, (sourcePair j).1 ≤ (sourcePair j).2) ∧
      ∃ n : ℕ+,
        (∑ j : Fin 2,
          ∑ m ∈ Set.Icc (sourcePair j).1 (sourcePair j).2, (m⁻¹ : ℚ)) = n := by
  constructor
  · intro j
    fin_cases j <;> decide
  · refine ⟨1, ?_⟩
    have h36 : Finset.Icc (3 : ℕ+) 6 = {3, 4, 5, 6} := by decide
    have h20 : Finset.Icc (20 : ℕ+) 20 = {20} := by decide
    have h3 : (3 : ℕ+) ∉ ({4, 5, 6} : Finset ℕ+) := by decide
    have h4 : (4 : ℕ+) ∉ ({5, 6} : Finset ℕ+) := by decide
    have h5 : (5 : ℕ+) ∉ ({6} : Finset ℕ+) := by decide
    simp only [sourcePair, Fin.sum_univ_two, if_pos, Fin.isValue]
    simp only [if_neg (by decide : (1 : Fin 2) ≠ 0)]
    rw [Set.toFinset_Icc, Set.toFinset_Icc]
    rw [h36, h20, Finset.sum_insert h3, Finset.sum_insert h4,
      Finset.sum_insert h5, Finset.sum_singleton, Finset.sum_singleton]
    change (3 : ℚ)⁻¹ + ((4 : ℚ)⁻¹ + ((5 : ℚ)⁻¹ + (6 : ℚ)⁻¹)) +
      (20 : ℚ)⁻¹ = 1
    norm_num

end Submissions.Erdos288SourceWitness.ExactArithmetic
```

- Canonical statement

```lean
import Mathlib.Data.PNat.Interval
import Mathlib.Data.Rat.Defs
import Mathlib.Algebra.BigOperators.Group.Finset.Basic

/-!
# The source witness for Erdős problem 288

The intervals `[3,6]` and `[20,20]` have reciprocal sums adding to one.
-/

namespace Statements.Erdos288SourceWitness

open scoped BigOperators

abbrev sourcePair : Fin 2 → ℕ+ × ℕ+ :=
  fun j => if j = 0 then (3, 6) else (20, 20)

abbrev statement : Prop :=
  (∀ j, (sourcePair j).1 ≤ (sourcePair j).2) ∧
    ∃ n : ℕ+,
      (∑ j : Fin 2,
        ∑ m ∈ Set.Icc (sourcePair j).1 (sourcePair j).2, (m⁻¹ : ℚ)) = n

theorem target : statement := sorry

end Statements.Erdos288SourceWitness
```

### 1. Only finitely many ordered pairs of nonempty intervals of positive integers have reciprocal sums whose total…

- Permalink: https://jig.so/p/48?s=1
- Status: open
- Filed: 2026-08-25T03:51:19.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Only finitely many ordered pairs of nonempty intervals of positive integers have reciprocal sums whose total is a positive integer.**

Direct formal-conjectures proposition after removing the yes/no answer wrapper. The exact published witness [3,6] and [20,20] proves the solution set nonempty. No Commons; computation is not part of the root verifier.

**Scope.**

Ordered pairs of inclusive positive-integer intervals with lower endpoint at most upper endpoint; exact rational reciprocal sums; positive-integer total.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.PNat.Interval
import Mathlib.Data.Rat.Defs
import Mathlib.Algebra.BigOperators.Group.Finset.Basic

/-!
# Erdős problem 288: integral sums over two reciprocal intervals

There are conjectured to be only finitely many ordered pairs of nonempty
intervals of positive integers whose reciprocal sums add to a positive integer.
-/

namespace Statements.Erdos288FiniteReciprocalIntervals

open scoped BigOperators

abbrev GoodPairs : Set (Fin 2 → ℕ+ × ℕ+) :=
  {I |
    ∀ j, (I j).1 ≤ (I j).2 ∧
      ∃ n : ℕ+,
        (∑ j : Fin 2,
          ∑ m ∈ Set.Icc (I j).1 (I j).2, (m⁻¹ : ℚ)) = n}

abbrev statement : Prop := Set.Finite GoodPairs

theorem target : statement := sorry

end Statements.Erdos288FiniteReciprocalIntervals
```

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