feat(ErdosProblems): 962

AUTHORS

@@ -19,6 +19,7 @@ Cong Lu
 Daniel Chin
 Eric Wieser
 Franz Huschenbeth
+Hansle Cho
 James Jordan
 Jean-Guillaume Durand
 Jofre Costa
@@ -30,10 +31,10 @@ Michael Rothgang
 Mirek Olšák
 Paul Lezeau
 Salvatore Mercuri
-Wojciech Nawrocki
 Seewoo Lee
 The bbchallenge Collaboration (bbchallenge.org)
 Thomas Hubert
+Wojciech Nawrocki
 Yan Yablonovskiy
 Yoh Tanimoto
 Yongxi Lin

FormalConjectures/ErdosProblems/962.lean

@@ -0,0 +1,91 @@
+/-
+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 962
+
+*References:*
+- [erdosproblems.com/962](https://www.erdosproblems.com/962)
+- [Er65] Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.
+- [Er76e] Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282.
+- [Tang](https://github.com/QuanyuTang/erdos-problem-962/blob/main/On_Erd%C5%91s_Problem_962.pdf)
+- [Tao](https://www.erdosproblems.com/forum/thread/962)
+-/
+
+open Classical Filter Real
+
+namespace Erdos962
+
+/--
+`Erdos962Prop n k` : there exists $m \le n$ such that each of
+$m+1, \ldots, m+k$ has a prime divisor strictly larger than $k$.
+-/
+def Erdos962Prop (n k : ℕ) : Prop :=
+  ∃ m ≤ n, ∀ i ∈ Set.Icc 1 k,
+    ∃ p : ℕ, Nat.Prime p ∧ k < p ∧ p ∣ (m + i)
+
+/--
+Let $k(n)$ be the maximal $k$ such that there exists $m \le n$ with
+$m+1, \ldots, m+k$ each divisible by a prime $> k$.
+-/
+noncomputable def k (n : ℕ) : ℕ :=
+  Nat.findGreatest (fun k => Erdos962Prop n k) n
+
+/--
+Main conjecture:
+
+$\log k(n) \le (\log n)^{(1/2 + o(1))}$
+-/
+@[category research open, AMS 11]
+theorem erdos_962 :
+    answer(sorry) ↔
+      ∃ ε : ℕ → ℝ,
+        (∀ δ > 0, ∀ᶠ n in atTop, |ε n| < δ) ∧
+        ∀ᶠ n : ℕ in atTop,
+          log (k n : ℝ)
+            ≤ rpow (log n) ((1 : ℝ) / 2 + ε n) := by
+  sorry
+
+/--
+Tang's lower bound [Tang]:
+
+$\log k(n) \ge (1/\sqrt{2} - o(1)) * \sqrt{\log n * \log \log n}$
+-/
+@[category research solved, AMS 11]
+theorem erdos_962.variants.tang_lower_bound :
+    ∃ ε : ℕ → ℝ,
+      (∀ δ > 0, ∀ᶠ n in atTop, |ε n| < δ) ∧
+      ∀ᶠ n : ℕ in atTop,
+          (1 / sqrt 2 - ε n) *
+              sqrt (log n * log (log n))
+            ≤ log (k n : ℝ) := by
+  sorry
+
+/--
+Tao's upper bound [Tao]:
+
+$k(n) \le (1 + o(1)) * n^{1/2}$
+-/
+@[category research solved, AMS 11]
+theorem erdos_962.variants.tao_upper_bound :
+    ∃ ε : ℕ → ℝ,
+      (∀ δ > 0, ∀ᶠ n in atTop, |ε n| < δ) ∧
+      ∀ᶠ n : ℕ in atTop,
+          (k n : ℝ) ≤ (1 + ε n) * sqrt n := by
+  sorry
+
+end Erdos962