FormalConjectures.ErdosProblems.«428»

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import FormalConjecturesUtil

Erdős Problem 428

Reference: erdosproblems.com/428


open Nat Filter Set


namespace Erdos428

The density ratio of set $A$ up to $n$ relative to the prime counting function $\pi(n)$.

noncomputable def primeDensityRatio (A : Set ℕ) (n : ℕ) : ℝ :=
  (A ∩ Icc 1 n).ncard / (primeCounting n)

Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\in A$ with $0 < a < n$ and $$\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?$$

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_428 :
    answer(sorry) ↔ ∃ A : Set ℕ,
      (∃ᶠ n in atTop, ∀ a ∈ A, 0 < a → a < n → (n - a).Prime) ∧
      liminf (fun n ↦ primeDensityRatio A n) atTop > 0 := by⊢ True ↔
  ∃ A,
    (∃ᶠ (n : ℕ) in atTop, ∀ a ∈ A, 0 < a → a < n → Nat.Prime (n - a)) ∧
      liminf (fun n => primeDensityRatio A n) atTop > 0
  sorryAll goals completed! 🐙


end Erdos428