FormalConjectures.ErdosProblems.«962»

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import FormalConjecturesUtil

Erdős Problem 962

References:

erdosproblems.com/962

[Er65] Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.

[Er76e] Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282.

Tang

Tao


open Classical Filter Real


namespace Erdos962

Erdos962Prop n k : there exists $m \le n$ such that each of $m+1, \ldots, m+k$ has a prime divisor strictly larger than $k$.

def Erdos962Prop (n k : ℕ) : Prop :=
  ∃ m ≤ n, ∀ i ∈ Set.Icc 1 k,
    ∃ p : ℕ, Nat.Prime p ∧ k < p ∧ p ∣ (m + i)

Let $k(n)$ be the maximal $k$ such that there exists $m \le n$ with $m+1, \ldots, m+k$ each divisible by a prime $> k$.

noncomputable def k (n : ℕ) : ℕ :=
  Nat.findGreatest (fun k => Erdos962Prop n k) n

Main conjecture:

$\log k(n) \le (\log n)^{(1/2 + o(1))}$

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_962 :
    answer(sorry) ↔
      ∃ ε : ℕ → ℝ,
        (∀ δ > 0, ∀ᶠ n in atTop, |ε n| < δ) ∧
        ∀ᶠ n : ℕ in atTop,
          log (k n : ℝ)
            ≤ rpow (log n) ((1 : ℝ) / 2 + ε n) := by⊢ True ↔ ∃ ε, (∀ δ > 0, ∀ᶠ (n : ℕ) in atTop, |ε n| < δ) ∧ ∀ᶠ (n : ℕ) in atTop, log ↑(k n) ≤ (log ↑n).rpow (1 / 2 + ε n)
  sorryAll goals completed! 🐙

Tang's lower bound [Tang]:

$\log k(n) \ge (1/\sqrt{2} - o(1)) * \sqrt{\log n * \log \log n}$

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_962.variants.tang_lower_bound :
    ∃ ε : ℕ → ℝ,
      (∀ δ > 0, ∀ᶠ n in atTop, |ε n| < δ) ∧
      ∀ᶠ n : ℕ in atTop,
          (1 / sqrt 2 - ε n) *
              sqrt (log n * log (log n))
            ≤ log (k n : ℝ) := by⊢ ∃ ε,
  (∀ δ > 0, ∀ᶠ (n : ℕ) in atTop, |ε n| < δ) ∧
    ∀ᶠ (n : ℕ) in atTop, (1 / √2 - ε n) * √(log ↑n * log (log ↑n)) ≤ log ↑(k n)
  sorryAll goals completed! 🐙

Tao's upper bound [Tao]:

$k(n) \le (1 + o(1)) * n^{1/2}$

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_962.variants.tao_upper_bound :
    ∃ ε : ℕ → ℝ,
      (∀ δ > 0, ∀ᶠ n in atTop, |ε n| < δ) ∧
      ∀ᶠ n : ℕ in atTop,
          (k n : ℝ) ≤ (1 + ε n) * sqrt n := by⊢ ∃ ε, (∀ δ > 0, ∀ᶠ (n : ℕ) in atTop, |ε n| < δ) ∧ ∀ᶠ (n : ℕ) in atTop, ↑(k n) ≤ (1 + ε n) * √↑n
  sorryAll goals completed! 🐙


end Erdos962