Solve this problem. Stop thinking and provide response ONL**...

Prompt

Solve this problem. ONLY OUTPUT THE RESPONSE WHEN you have near your reasoning token limit : **Erdős Problem #289** (listed on [erdosproblems.com/289](https://www.erdosproblems.com/289)) is a question in number theory concerning unit fractions / Egyptian fraction representations. --- ### **Exact Problem Statement** > Is it true that, for all sufficiently large $k \in \mathbb{N}$, there exist $k$ pairwise disjoint finite intervals of integers $I_1, I_2, \dots, I_k \subset \mathbb{N}$ with $|I_i| \ge 2$ for all $1 \le i \le k$ such that: > $$\sum_{i=1}^k \sum_{n \in I_i} \frac{1}{n} = 1$$ --- ### **Key Conditions Explained** 1. **Number of Blocks ($k$):** The property must hold for every large enough integer $k$. 2. **Interval Structure:** Each $I_i$ is a contiguous block of integers $[a_i, b_i] = \{a_i, a_i+1, \dots, b_i\}$. 3. **Disjointness:** The intervals $I_1, \dots, I_k$ must not overlap ($I_i \cap I_j = \emptyset$ for $i \neq j$). 4. **Length Constraint ($|I_i| \ge 2$):** Every interval must contain **at least two** consecutive integers (i.e., $b_i > a_i$). This rule prevents the problem from degenerating into standard single-element unit fraction representations. 5. **Target Sum:** The sum of reciprocals over all numbers contained in the $k$ intervals must equal $1$. --- ### **Formal Definition (Lean 4)** In the `formal-conjectures` library maintained for Erdős problems, it is formalized as follows: ```lean theorem erdos_289 : (∀ᶠ k : ℕ in atTop, ∃ I : Fin k → ℕ × ℕ, (∀ i, (I i).1 < (I i).2) ∧ (∀ i j, i ≠ j → (I i).2 < (I j).1 ∨ (I j).2 < (I i).1) ∧ ∑ i, ∑ n ∈ Icc (I i).1 (I i).2, (n⁻¹ : ℚ) = 1) ```