/-
Copyright 2025 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 FormalConjecturesUtilErdős Problem 370
namespace Erdos370
Are there infinitely many $n$ such that the largest prime factor of $n$ is $< n^{\frac{1}{2}}$ and the largest prime factor of $n + 1$ is $< (n + 1)^{\frac{1}{2}}$.
Steinerberger has pointed out this problem has a trivial solution.
This was formalized in Lean by Alexeev using Aristotle.
@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos370.lean",
formal_proof using formal_conjectures at
"https://github.com/XC0R/formal-conjectures/blob/f58dea7d2cc5c9da2e050ec80a73e838b54a6dd2/FormalConjectures/ErdosProblems/370.lean#L73"]
theorem erdos_370 : answer(True) ↔
{ n | Nat.maxPrimeFac n < √n ∧ Nat.maxPrimeFac (n + 1) < √(n + 1) }.Infinite := ⊢ True ↔ {n | ↑n.maxPrimeFac < √↑n ∧ ↑(n + 1).maxPrimeFac < √(↑n + 1)}.Infinite
All goals completed! 🐙
end Erdos370