FormalConjectures.ErdosProblems.«4»

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import FormalConjecturesUtil

Erdős Problem 4

Reference: erdosproblems.com/4


open Real


namespace Erdos4


def Erdos4For (C : ℝ) : Prop :=
  {n : ℕ | (n + 1).nth Nat.Prime - n.nth Nat.Prime >
    C * log (log n) * log (log (log (log n))) / (log (log (log n))) ^ 2 * log n}.Infinite

Is it true that, for any $C > 0$, there infinitely many $n$ such that: $$ p_{n + 1} - p_n > C \frac{\log\log n\log\log\log\log n}{(\log\log\log n) ^ 2}\log n $$

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_4 : answer(True) ↔ (∀ C > 0, Erdos4For C) := by⊢ True ↔ ∀ C > 0, Erdos4For C
  sorryAll goals completed! 🐙

Rankin's theorem: there exists a positive constant $C$ such that Erdos4For C holds.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_4.variants.rankin :
    ∃ C > 0, Erdos4For C := by⊢ ∃ C > 0, Erdos4For C
  sorryAll goals completed! 🐙


end Erdos4