FormalConjectures.ErdosProblems.«452»

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import FormalConjecturesUtil

Erdős Problem 452

Reference: erdosproblems.com/452

namespace Erdos452open scoped ArithmeticFunction.omega

The greatest length of an interval in $[x,2x]$ on which $\omega(n) > \log\log n$ everywhere.

noncomputable def largeOmegaIntervalLength (x : ℕ) : ℕ := byx:ℕ⊢ ℕ
  classical
  exact Nat.findGreatest
    (fun len ↦ ∃ start, x ≤ start ∧ start + len ≤ 2 * x + 1 ∧
      ∀ n ∈ Finset.Ico start (start + len),
        Real.log (Real.log (n : ℝ)) < (ω n : ℝ))
    (x + 1)All goals completed! 🐙

Determine the largest length of an interval in $[x,2x]$ on which $\omega(n) > \log\log n$ everywhere.

@[category research open, AMS 11]
theorem erdos_452 : largeOmegaIntervalLength = answer(sorry) := by⊢ largeOmegaIntervalLength = sorry
  sorryAll goals completed! 🐙

end Erdos452