FormalConjectures.ErdosProblems.«885»

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import FormalConjecturesUtil

Erdős Problem 885

References:

erdosproblems.com/885

[ErRo97] Erdős, P. and Rosenfeld, M., The factor-difference set of integers. (1997)

[Ji99] Jiménez-Urroz, J., A note on a conjecture of Erdős and {R}osenfeld. (1999)

[Br19] Bremner, A., On a problem of Erdős related to common factor differences. (2019)


open Nat Set Finset


namespace Erdos885

For integer $n \geq 1$ we define the factor difference set of $n$ by $D(n) = {|a-b| : n=ab}$.

def factorDifferenceSet (n : ℕ) : Set ℕ :=
  {d | ∃ a b : ℕ, n = a * b ∧ (d : ℤ) = |(a : ℤ) - b|}

Is it true that, for every $k \geq 1$, there exist integers $N_1 < \dots < N_k$ such that $|\cap_i D(N_i)| \geq k$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_885 : answer(sorry) ↔ ∀ k ≥ 1,
    ∃ Ns : Finset ℕ,
      (∀ n ∈ Ns, 1 ≤ n) ∧
      Ns.card = k ∧
      (⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ k := by⊢ True ↔ ∀ k ≥ 1, ∃ Ns, (∀ n ∈ Ns, 1 ≤ n) ∧ #Ns = k ∧ (⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ k
  sorryAll goals completed! 🐙

Erdős and Rosenfeld [ErRo97] proved this is true for $k=2$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_885.variants.k_eq_2 :
    ∃ Ns : Finset ℕ,
      (∀ n ∈ Ns, 1 ≤ n) ∧
      Ns.card = 2 ∧
      (⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ 2 := by⊢ ∃ Ns, (∀ n ∈ Ns, 1 ≤ n) ∧ #Ns = 2 ∧ (⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ 2
  sorryAll goals completed! 🐙

Jiménez-Urroz [Ji99] proved this for $k=3$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_885.variants.k_eq_3 :
    ∃ Ns : Finset ℕ,
      (∀ n ∈ Ns, 1 ≤ n) ∧
      Ns.card = 3 ∧
      (⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ 3 := by⊢ ∃ Ns, (∀ n ∈ Ns, 1 ≤ n) ∧ #Ns = 3 ∧ (⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ 3
  sorryAll goals completed! 🐙

Bremner [Br19] proved this for $k=4$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_885.variants.k_eq_4 :
    ∃ Ns : Finset ℕ,
      (∀ n ∈ Ns, 1 ≤ n) ∧
      Ns.card = 4 ∧
      (⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ 4 := by⊢ ∃ Ns, (∀ n ∈ Ns, 1 ≤ n) ∧ #Ns = 4 ∧ (⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ 4
  sorryAll goals completed! 🐙


end Erdos885