| 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 |
|
|
