FormalConjectures.ErdosProblems.«208»

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import FormalConjecturesUtil

Erdős Problem 208

Reference: erdosproblems.com/208


open Filter Real


namespace Erdos208

The sequence of squarefree numbers, denoted by s as in Erdős problem 208.

noncomputable def erdos208.s : ℕ → ℕ := Nat.nth Squarefree


open erdos208

Let $s_1 < s_2 < \dots$ be the sequence of squarefree numbers. Is it true that for any $\epsilon > 0$ and large $n$, $s_{n+1} - s_n \ll_\epsilon s_n^\epsilon$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_208.parts.i : answer(sorry) ↔
    ∀ ε > (0 : ℝ), (fun n => (s (n + 1) - s n : ℝ)) =O[atTop] (fun n => (s n : ℝ)^ε) := by⊢ True ↔ ∀ ε > 0, (fun n => ↑(s (n + 1)) - ↑(s n)) =O[atTop] fun n => ↑(s n) ^ ε sorryAll goals completed! 🐙

Let $s_1 < s_2 < \dots$ be the sequence of squarefree numbers. Is it true that $s_{n + 1} - s_n \le (1 + o(1)) \cdot (\pi^2 / 6) \cdot \log (s_n) / \log (\log (s_n))$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_208.parts.ii : answer(sorry) ↔ ∃ (c : ℕ → ℝ), (c =o[atTop] (1 : ℕ → ℝ)) ∧ ∀ᶠ n in atTop,
      s (n + 1) - s n ≤ (1 + (c n)) * (π^2 / 6) * log (s n) / log (log (s n)) := by⊢ True ↔
  ∃ c,
    c =o[atTop] 1 ∧ ∀ᶠ (n : ℕ) in atTop, ↑(s (n + 1)) - ↑(s n) ≤ (1 + c n) * (π ^ 2 / 6) * log ↑(s n) / log (log ↑(s n))
  sorryAll goals completed! 🐙

In [Er79] Erdős says perhaps $s_{n+1} - s_n \ll \log s_n$, but he is 'very doubtful'.

[Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_208.variants.log_bound :
    (fun n ↦ (s (n + 1) - s n : ℝ)) =O[atTop] fun n ↦ log (s n) := by⊢ (fun n => ↑(s (n + 1)) - ↑(s n)) =O[atTop] fun n => log ↑(s n) sorryAll goals completed! 🐙


end Erdos208