FormalConjectures.ErdosProblems.«363»

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import FormalConjecturesUtil

Erdős Problem 363

References:

erdosproblems.com/363

[BaBe07] Bauer, Mark and Bennett, Michael A., On a question of Erd\H{o}s and Graham. Enseign. Math. (2) (2007), 259--264.

[BeVL12] Bennett, Michael A. and Van Luijk, Ronald, Squares from blocks of consecutive integers: a problem of Erd\H{o}s and Graham. Indag. Math. (N.S.) (2012), 123--127.

[Ul05] Ulas, Maciej, On products of disjoint blocks of consecutive integers. Enseign. Math. (2) (2005), 331--334.


namespace Erdos363


open Finset

A finite set of naturals is an interval of naturals.

def IsInterval (I : Finset ℕ) : Prop :=
  ∃ a b : ℕ, I = Icc a b

A collection of intervals as in Erdős Problem 363.

def IsValidCollection (S : List (Finset ℕ)) : Prop :=
  (∀ I ∈ S, IsInterval I) ∧
  (∀ I ∈ S, 4 ≤ I.card) ∧
  S.Pairwise Disjoint ∧
  IsSquare ((S.map (fun I => ∏ m ∈ I, m)).prod)

Is it true that there are only finitely many collections of disjoint intervals $I_1,\ldots,I_n$ of size $\lvert I_i\rvert \geq 4$ for $1\leq i\leq n$ such that$$\prod_{1\leq i\leq n}\prod_{m\in I_i}m$$is a square?

This is false: Ulas [Ul05] constructed infinitely many such collections.

@[category research solved, AMS 11,
  formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos363.lean"]
theorem declaration uses 'sorry'erdos_363 : answer(False) ↔ {S : List (Finset ℕ) | IsValidCollection S}.Finite := by⊢ False ↔ {S | IsValidCollection S}.Finite
  sorryAll goals completed! 🐙


end Erdos363