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module
public import FormalConjecturesUtilErdős Problem 289
Reference: erdosproblems.com/289
@[expose] public sectionopen Asymptotics Filter Finsetnamespace Erdos289Is it true that, for all sufficiently large $k$, there exist finite intervals $I_1, \dotsc, I_k \subset \mathbb{N}$, distinct, not overlapping or adjacent, with $|I_i| \geq 2$ for $1 \leq i \leq k$ such that $$ 1 = \sum_{i=1}^k \sum_{n \in I_i} \frac{1}{n}? $$ Here two intervals are adjacent if their union is again an interval, so any two of the $I_i$ must be separated by at least one integer.
@[category research open, AMS 11]
theorem erdos_289 : answer(sorry) ↔
(∀ᶠ k : ℕ in atTop, ∃ I : Fin k → ℕ × ℕ,
(∀ i, (I i).1 < (I i).2) ∧
(∀ i j, i ≠ j → (I i).2 + 1 < (I j).1 ∨ (I j).2 + 1 < (I i).1) ∧
∑ i, ∑ n ∈ .Icc (I i).1 (I i).2, (n⁻¹ : ℚ) = 1) := ⊢ True ↔
∀ᶠ (k : ℕ) in atTop,
∃ I,
(∀ (i : Fin k), (I i).1 < (I i).2) ∧
(∀ (i j : Fin k), i ≠ j → (I i).2 + 1 < (I j).1 ∨ (I j).2 + 1 < (I i).1) ∧
∑ i, ∑ n ∈ Icc (I i).1 (I i).2, (↑n)⁻¹ = 1
All goals completed! 🐙end Erdos289