FormalConjectures.ErdosProblems.«288»

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import FormalConjecturesUtil

Erdős Problem 288

Reference: erdosproblems.com/288


namespace Erdos288

Is it true that there are only finitely many pairs of intervals $I_1$, $I_2$ such that $$ \sum_{n_1 \in I_1} \frac{1}{n_1} + \sum_{n_2 \in I_2} \frac{1}{n_2} \in \mathbb{N}? $$

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_288 : answer(sorry) ↔ Set.Finite { I : Fin 2 → ℕ+ × ℕ+ |
    ∀ j, (I j).1 ≤ (I j).2 ∧
      ∃ n : ℕ+, (∑ j : Fin 2, ∑ nⱼ ∈ Set.Icc (I j).1 (I j).2, (nⱼ⁻¹ : ℚ)) = n } := by⊢ True ↔
  {I | ∀ (j : Fin 2), (I j).1 ≤ (I j).2 ∧ ∃ n, ∑ j, ∑ nⱼ ∈ (Set.Icc (I j).1 (I j).2).toFinset, (↑↑nⱼ)⁻¹ = ↑↑n}.Finite
  sorryAll goals completed! 🐙

This is still open even if $|I_2| = 1$.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_288.variants.i2_card_eq_1 : answer(sorry) ↔ Set.Finite { (I, n₂) : (ℕ+ × ℕ+) × ℕ+ |
    I.1 ≤ I.2 ∧ ∃ n : ℕ+, ∑ n₁ ∈ Set.Icc I.1 I.2, (n₁⁻¹ : ℚ) + (n₂⁻¹ : ℚ) = n } := by⊢ True ↔ {(I, n₂) | I.1 ≤ I.2 ∧ ∃ n, ∑ n₁ ∈ (Set.Icc I.1 I.2).toFinset, (↑↑n₁)⁻¹ + (↑↑n₂)⁻¹ = ↑↑n}.Finite
  sorryAll goals completed! 🐙

It is perhaps true with two intervals replaced by any $k$ intervals.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_288.variants.k_intervals : answer(sorry) ↔ ∀ k, Set.Finite { I : Fin k → ℕ+ × ℕ+ |
    ∀ j, (I j).1 ≤ (I j).2 ∧
      ∃ n : ℕ+, (∑ j : Fin k, ∑ nⱼ ∈ Set.Icc (I j).1 (I j).2, (nⱼ⁻¹ : ℚ)) = n } := by⊢ True ↔
  ∀ (k : ℕ),
    {I | ∀ (j : Fin k), (I j).1 ≤ (I j).2 ∧ ∃ n, ∑ j, ∑ nⱼ ∈ (Set.Icc (I j).1 (I j).2).toFinset, (↑↑nⱼ)⁻¹ = ↑↑n}.Finite
  sorryAll goals completed! 🐙

Is it true for any $k > 2$ that only finitely many $k$ intervals satisfy this condition?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_288.variants.exists_k_gt_2 : answer(sorry) ↔
    ∃ k > 2, Set.Finite { I : Fin k → ℕ+ × ℕ+ |
      ∀ j, (I j).1 ≤ (I j).2 ∧
        ∃ n : ℕ+, (∑ j : Fin k, ∑ nⱼ ∈ Set.Icc (I j).1 (I j).2, (nⱼ⁻¹ : ℚ)) = n } := by⊢ True ↔
  ∃ k > 2,
    {I | ∀ (j : Fin k), (I j).1 ≤ (I j).2 ∧ ∃ n, ∑ j, ∑ nⱼ ∈ (Set.Icc (I j).1 (I j).2).toFinset, (↑↑nⱼ)⁻¹ = ↑↑n}.Finite
  sorryAll goals completed! 🐙


end Erdos288