FormalConjectures.ErdosProblems.«646»

/-
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 FormalConjecturesUtil

Erdős Problem 646

References:

erdosproblems.com/646

[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

[Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.

[Be97] Berend, Daniel, On the parity of exponents in the factorization of $n!$. J. Number Theory (1997), 13-19.

open scoped Natnamespace Erdos646

Let $p_1,\ldots,p_k$ be distinct primes. Are there infinitely many $n$ such that $n!$ is divisible by an even power of each of the $p_i$?

The answer is yes, proved by Berend [Be97], who further proved that the sequence of such $n$ has bounded gaps (where the bound depends on the initial set of primes).

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos646.lean"]
theorem erdos_646 : answer(True) ↔
    ∀ S : Finset ℕ, (∀ p ∈ S, p.Prime) →
      {n : ℕ | ∀ p ∈ S, Even (padicValNat p (n !))}.Infinite := by⊢ True ↔ ∀ (S : Finset ℕ), (∀ p ∈ S, Nat.Prime p) → {n | ∀ p ∈ S, Even (padicValNat p n !)}.Infinite
  sorryAll goals completed! 🐙

end Erdos646