/-
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 FormalConjecturesUtil
namespace Erdos137
Let $k\geq 3$. Can the product of any $k$ consecutive integers $N$ ever be powerful? That is, must there always exist a prime $p\mid N$ such that $p^2\nmid N$?
@[category research open, AMS 11]
theorem erdos_137 : answer(sorry) ↔ ∀ k ≥ 3, ∀ n, ¬ (∏ x ∈ Finset.Ioc n (n + k), x).Powerful := ⊢ True ↔ ∀ k ≥ 3, ∀ (n : ℕ), ¬(∏ x ∈ Finset.Ioc n (n + k), x).Powerful
All goals completed! 🐙
Let $k\geq 2$. Erdős and Selfridge [ES75] proved that the product of any $k$ consecutive integers $N$ cannot be a perfect power.
[ES75] P. Erdös, J. L. Selfridge, "The product of consecutive integers is never a power", Illinois J. Math. 19(2): 292-301, 1975
@[category research solved, AMS 11]
theorem erdos_137.variants.perfect_power (k : ℕ) (hk : k ≥ 2) (n : ℕ) (x l : ℕ) (hl : 2 ≤ l) :
(∏ x ∈ Finset.Ioc n (n + k), x) ≠ x ^ l := k:ℕhk:k ≥ 2n:ℕx:ℕl:ℕhl:2 ≤ l⊢ ∏ x ∈ Finset.Ioc n (n + k), x ≠ x ^ l
All goals completed! 🐙
Erdős [Er82c] conjectures that, if $k$ is fixed, then for all $n$ sufficiently large and all positive integers $m$, there must be at least $k$ distinct primes $p$ such that $p\mid m(m+1)\cdots (m+n)$ and yet $p^2$ does not divide the right hand side.
[Er82c] Erdős, Paul, "Miscellaneous problems in number theory". Congr. Numer. (1982), 25-45.,
@[category research open, AMS 11]
theorem erdos_137.variants.multiple_powerful_factors (k : ℕ) : ∀ᶠ n in Filter.atTop,
∀ (m : ℕ) (hm : 0 < m),
letI N := ∏ x ∈ Finset.Ioc m (m + n), x
∃ P : Finset ℕ, P.card = k ∧ ∀ p ∈ P, p.Prime ∧
p ∣ N ∧ ¬ p ^ 2 ∣ N := k:ℕ⊢ ∀ᶠ (n : ℕ) in Filter.atTop,
∀ (m : ℕ),
0 < m →
∃ P,
P.card = k ∧ ∀ p ∈ P, Nat.Prime p ∧ p ∣ ∏ x ∈ Finset.Ioc m (m + n), x ∧ ¬p ^ 2 ∣ ∏ x ∈ Finset.Ioc m (m + n), x
All goals completed! 🐙
end Erdos137