1. FormalConjectures
  2. OEIS
  3. «80170»
/- 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 import Mathlib.Algebra.IsPrimePow

Conjecture relating two characterizations of a set of integers.

Informal Statement: For an integer $k \ge 2$, the following are equivalent:

    The greatest common divisor of the binomial coefficients $\binom{2k}{k}, \binom{3k}{k}, \dots, \binom{(k+1)k}{k} = 1$.

    Writing prime factorization of $k$ as $k = \prod p_i^{e_i}$, and let $P = \max_i p_i^{e_i}$, one has $k / P > P$.

This conjecture asserts that the sequence defined by 1. is obtained by taking 1 off each number in the sequence defined by 2.

References:

namespace OeisA80170

The gcd of the binomial coefficients $\binom{2k}{k}, \binom{3k}{k}, \dots, \binom{(k+1)k}{k} = 1$.

def A (k : ℕ) : Prop := (Finset.range k).gcd (fun i => Nat.choose ((i + 2) * k) k) = 1

Let $P$ be the largest prime power dividing k. Then $k / P > P$.

def B (k : ℕ) : Prop := let P := ((Nat.divisors k).filter IsPrimePow).max.getD 0 k / P > P @[category test, AMS 11] theorem a_2 : ¬A 2 := ⊢ ¬A 2 ⊢ ¬((Finset.range 2).gcd fun i => ((i + 2) * 2).choose 2) = 1; All goals completed! 🐙 @[category test, AMS 11] theorem a_3 : ¬A 3 := ⊢ ¬A 3 ⊢ ¬((Finset.range 3).gcd fun i => ((i + 2) * 3).choose 3) = 1; All goals completed! 🐙 @[category test, AMS 11] theorem a_4 : ¬A 4 := ⊢ ¬A 4 ⊢ ¬((Finset.range 4).gcd fun i => ((i + 2) * 4).choose 4) = 1; All goals completed! 🐙 @[category test, AMS 11] theorem a_5 : ¬A 5 := ⊢ ¬A 5 ⊢ ¬((Finset.range 5).gcd fun i => ((i + 2) * 5).choose 5) = 1; All goals completed! 🐙 @[category test, AMS 11] theorem a_6 : ¬A 6 := ⊢ ¬A 6 ⊢ ¬((Finset.range 6).gcd fun i => ((i + 2) * 6).choose 6) = 1; All goals completed! 🐙

Conjecture: The gcd condition is equivalent to the prime power condition. This has been conjectured by Ralf Stephan.

Both the natural-language proof and its Lean 4 formalization were carried out by the KLMM MechMath Agent Team; see the formal_proof attribute.

References:

@[category research solved, AMS 11, formal_proof using formal_conjectures at "https://github.com/guodk/formal-conjectures/blob/0720658844d76a50d48e4baa152eef14d4462907/FormalConjectures/OEIS/80170.lean#L1823"] theorem declaration uses 'sorry'gcdCondition_iff_primePowerCondition (k : ℕ) (hk : 2 ≤ k) : A k ↔ B (k + 1) := k:ℕhk:2 ≤ k⊢ A k ↔ B (k + 1) All goals completed! 🐙 end OeisA80170