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b797a5b | 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 | import OAI.NumberTheory.PrimeGaps.Density
import OAI.NumberTheory.PrimeGaps.RatioAlgebra
namespace OAI
namespace Problem344
theorem mem_ratioIncreaseIndices_iff_gap {n : ℕ} :
n ∈ ratioIncreaseIndices ↔
1 ≤ n ∧ (primeAt n : ℝ) / (n : ℝ) <
(primeAt (n + 1) : ℝ) - (primeAt n : ℝ) := by
constructor
· rintro ⟨hn, hratio⟩
exact ⟨hn, (div_lt_div_succ_iff_gap hn).1 hratio⟩
· rintro ⟨hn, hgap⟩
exact ⟨hn, (div_lt_div_succ_iff_gap hn).2 hgap⟩
theorem mem_ratioIncreaseIndices_of_mem_largeGapIndices {C : ℝ} {n : ℕ}
(hbound : (primeAt n : ℝ) / (n : ℝ) ≤ C * Real.log (primeAt n : ℝ))
(hgap : n ∈ largeGapIndices C) : n ∈ ratioIncreaseIndices := by
apply mem_ratioIncreaseIndices_iff_gap.2
exact ⟨hgap.1, lt_of_le_of_lt hbound hgap.2⟩
theorem eventually_largeGapIndices_subset_ratioIncreaseIndices {C : ℝ}
(hbound : ∃ N0 : ℕ, ∀ n : ℕ, N0 ≤ n →
(primeAt n : ℝ) / (n : ℝ) ≤ C * Real.log (primeAt n : ℝ)) :
∃ N0 : ℕ, ∀ n : ℕ, N0 ≤ n →
n ∈ largeGapIndices C → n ∈ ratioIncreaseIndices := by
obtain ⟨N0, hN0⟩ := hbound
exact ⟨N0, fun n hn => mem_ratioIncreaseIndices_of_mem_largeGapIndices (hN0 n hn)⟩
end Problem344
end OAI
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