FormalConjectures.GreensOpenProblems.«62»

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import FormalConjecturesUtil

Ben Green's Open Problem 62

Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$. Is every $x \in {1, \ldots, p-1}$ congruent to some product $a_1 a_2$ where $a_1, a_2 \in A$?

This is a problem of Erdős, Odlyzko, and Sárközy [105] from 1987.

Reference: Ben Green's Open Problem 62

open Filternamespace Green62

Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$. Is every $x \in {1, \ldots, p-1}$ congruent to some product $a_1 a_2$ where $a_1, a_2 \in A$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'green_62 :
    answer(sorry) ↔
      ∀ᶠ p in atTop, p.Prime →
        let A := (Finset.range p).filter Nat.Prime
        ∀ x : ℕ, 1 ≤ x ∧ x < p →
          ∃ a₁ ∈ A, ∃ a₂ ∈ A, (x : ZMod p) = (a₁ * a₂ : ZMod p) := by⊢ True ↔
  ∀ᶠ (p : ℕ) in atTop,
    Nat.Prime p →
      let A := Finset.filter Nat.Prime (Finset.range p);
      ∀ (x : ℕ), 1 ≤ x ∧ x < p → ∃ a₁ ∈ A, ∃ a₂ ∈ A, ↑x = ↑a₁ * ↑a₂
  sorryAll goals completed! 🐙


end Green62