FormalConjectures.ErdosProblems.«488»

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import FormalConjecturesUtil

Erdős Problem 488

Reference: erdosproblems.com/488


open Classical


namespace Erdos488

Let $A$ be a finite set and $$B={ n \geq 1 : a\mid n\textrm{ for some }a\in A}.$$ Is it true that, for every $m>n\geq \max(A)$, $$\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?$$

@[category research open, AMS 5 11]
theorem declaration uses 'sorry'erdos_488 : answer(sorry) ↔ ∀ (A : Finset ℕ), A.Nonempty →
    -- These are needed for the reasons outlined here: https://github.com/google-deepmind/formal-conjectures/pull/256
    0 ∉ A → 1 ∉ A →
    letI B := {n ≥ 1 | ∃ a ∈ A, a ∣ n}
    ∀ᵉ (n : ℕ) (m > n), A.max ≤ n →
      ((Finset.Icc 1 m).filter (· ∈ B)).card / (m : ℚ) <
        2 * ((Finset.Icc 1 n).filter (· ∈ B)).card / n := by⊢ True ↔
  ∀ (A : Finset ℕ),
    A.Nonempty →
      0 ∉ A →
        1 ∉ A →
          ∀ (n m : ℕ),
            m > n →
              A.max ≤ ↑n →
                ↑{x ∈ Finset.Icc 1 m | x ∈ {n | n ≥ 1 ∧ ∃ a ∈ A, a ∣ n}}.card / ↑m <
                  2 * ↑{x ∈ Finset.Icc 1 n | x ∈ {n | n ≥ 1 ∧ ∃ a ∈ A, a ∣ n}}.card / ↑n
  sorryAll goals completed! 🐙


end Erdos488