Erdős Problem 4

Reference: erdosproblems.com/4

def Erdos4.Erdos4For (C : ℝ) : Prop

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theorem Erdos4.erdos_4 : (∀ C > 0, Erdos4For C) ↔ True

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

theorem Erdos4.erdos_4.variants.rankin : ∃ C > 0, Erdos4For C