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465b10d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 | import Mathlib
namespace OAI
namespace LargePrimeGaps
noncomputable def prime (n : ℕ) : ℕ := Nat.nth Nat.Prime (n - 1)
noncomputable def largeGapIndices (C : ℝ) (N : ℕ) : Finset ℕ := by
classical
exact (Finset.Icc 1 N).filter fun n =>
C * Real.log (prime n : ℝ) < (prime (n + 1) : ℝ) - (prime n : ℝ)
end LargePrimeGaps
noncomputable section
namespace Problem344
def primeAt (n : ℕ) : ℕ :=
Nat.nth Nat.Prime (n - 1)
def initialCount (A : Set ℕ) (N : ℕ) : ℕ := by
classical
exact ((Finset.Icc 1 N).filter (fun n => n ∈ A)).card
def lowerAsymptoticDensity (A : Set ℕ) : ℝ :=
sSup {d : ℝ | ∃ N0 : ℕ, ∀ N : ℕ, N0 ≤ N →
d ≤ (initialCount A N : ℝ) / (N : ℝ)}
def ratioIncreaseIndices : Set ℕ :=
{n | 1 ≤ n ∧
(primeAt n : ℝ) / (n : ℝ) <
(primeAt (n + 1) : ℝ) / ((n + 1 : ℕ) : ℝ)}
theorem large_gaps_and_ratio_density :
(∀ C : ℝ, 0 < C →
∃ c : ℝ, 0 < c ∧ ∃ N₀ : ℕ, ∀ N : ℕ, N₀ ≤ N →
c * (N : ℝ) ≤ ((LargePrimeGaps.largeGapIndices C N).card : ℝ)) ∧
0 < lowerAsymptoticDensity ratioIncreaseIndices := by
sorry
end Problem344
end
end OAI
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