diff --git a/AUTHORS b/AUTHORS index ee97b877c3..3a629100a2 100644 --- a/AUTHORS +++ b/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 diff --git a/FormalConjectures/ErdosProblems/962.lean b/FormalConjectures/ErdosProblems/962.lean new file mode 100644 index 0000000000..e92a16f1a4 --- /dev/null +++ b/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