feat(ErdosProblems): 1139

FormalConjectures/ErdosProblems/1139.lean

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+/-
+Copyright 2026 The Formal Conjectures Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    https://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+-/
+
+import FormalConjectures.Util.ProblemImports
+/-!
+# Erdős Problem 1139
+
+*Reference:* [erdosproblems.com/1139](https://www.erdosproblems.com/1139)
+-/
+
+open Nat Filter
+open scoped ArithmeticFunction
+open scoped Topology
+
+namespace Erdos1139
+
+/--
+Let $1\leq u_1 < u_2 < \cdots$ be the sequence of integers with at most $2$ prime factors.
+Is it true that \[\limsup_{k \to \infty} \frac{u_{k+1}-u_k}{\log k}=\infty?\]
+-/
+@[category research open, AMS 11]
+theorem erdos_1139 :
+    answer(sorry) ↔
+      let u := Nat.nth (fun n ↦ 0 < n ∧ Ω n ≤ 2)
+      atTop.limsup (fun k : ℕ ↦ (((u (k + 1) : ℝ) - (u k : ℝ)) / Real.log ↑k : EReal)) = ⊤ := by
+  sorry
+
+end Erdos1139