{"id":"def:mills","label":"mills","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A positive real number \\(x\\) is Mills if \\(1 {\\lt} x\\) and for all positive integers \\(n\\), the number \\(\\lfloor x^{3^n} \\rfloor \\) is prime.","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Mills","latex_link":"sect0001.html#def:mills","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Mills","lean_decl_name":"Mills","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Mills","source_path":"./Mills/Basic copy.lean","lean4_code":"def Mills (x : ℝ) : Prop := 1 < x ∧ ∀ k : ℕ, 0 < k → Nat.Prime (Nat.floor (x ^ (3 ^ k)))\n","line":121,"end_line":121,"proof_status":"statement_formalized"}
{"id":"prop:mills","label":"mills","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":["def:mills"],"nlp_text":"There exists a Mills number.","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/exists_Mills","latex_link":"sect0001.html#prop:mills","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/exists_Mills","lean_decl_name":"exists_Mills","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Mills","source_path":"./Mills/Basic.lean","lean4_code":"theorem exists_Mills : ∃ A : ℝ, Mills A := by\n  have : 1 < pp 1 := by rw [pp]; linarith [(pp' 1).property, BHP_const_nat_ge2]\n  use (left_lim pp)\n  constructor\n  · exact (by linarith [(left_gt_1 pp this), (left_le_sup pp (fun n ↦ aux' n) 1)])\n  · intro k h\n    rw [left_floor_eq_seq pp (fun n ↦ (hpp n).left) (fun n ↦ aux' n) k, ← Nat.succ_pred_eq_of_pos h]\n    exact (hpp k.pred).right.right.right\n    exact h\n\n","line":409,"end_line":418,"proof_status":"statement_formalized"}
{"id":"prop:proposition4","label":"proposition4","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(W\\) has minimum.","doc_url":null,"latex_link":"sect0001.html#prop:proposition4","lean_link":null,"lean_decl_name":"proposition4","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:lemma6","label":"lemma6","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all natural numbers \\(k\\), we have \\[ p_k \\leq p_{k+1} {\\lt} (p_k ^3 + 1) ^3 - 1. \\]","doc_url":null,"latex_link":"sect0001.html#lem:lemma6","lean_link":null,"lean_decl_name":"lemma6","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"prop:proposition5","label":"proposition5","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":["lem:lemma6","thm:BHP"],"nlp_text":"\\(\\theta = 21/40\\). There exists a natural number \\(k_0\\) such that for all \\(k \\geq k_0\\), we have \\[ p_k ^3 \\leq p_{k+1} \\leq p_k ^3 + p_k ^{3\\theta }. \\]","doc_url":null,"latex_link":"sect0001.html#prop:proposition5","lean_link":null,"lean_decl_name":"proposition5","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:lemma7","label":"lemma7","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":["prop:proposition5"],"nlp_text":"There exists \\(\\gamma {\\gt} 0\\) and \\(k_1 {\\gt} 1\\) such that for all \\(k \\geq k_1\\), we have \\[ \\lvert A ^{3 ^k} - p_k\\rvert \\leq e^{-3 ^k\\gamma }. \\]","doc_url":null,"latex_link":"sect0001.html#lem:lemma7","lean_link":null,"lean_decl_name":"lemma7","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:mills_irrational","label":"mills_irrational","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":["lem:lemma7"],"nlp_text":"The Mills’ constant is irrational.","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Mills_irrational","latex_link":"sect0001.html#thm:mills_irrational","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Mills_irrational","lean_decl_name":"Mills_irrational","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Mills","source_path":"./Mills/Basic copy.lean","lean4_code":"theorem Mills_irrational : Irrational A := by\n  rw [Irrational]; simp\n  intro r h\n  have h₁ : 1 < r := by\n    have : Rat.cast 1  < (r : ℝ) := by rw [h]; simp; exact Mills_exists.left\n    apply Rat.cast_lt.1 this\n  have h₂ : ∀ n : ℤ, ↑n ≠ r := by\n    intro n h₃\n    rcases (nat_pos r h₁ n h₃) with ⟨m, hm⟩\n    rw [← h₃, ← hm] at h\n    apply A_not_nat m h.symm\n  rcases lem7 with ⟨γ, _, k₁, _, h₃⟩\n  rcases Mahler r γ h₁ h₂ with ⟨K, h₄⟩\n  let k := max K k₁\n  have h₅ : K ≤ 3 ^ k := by\n    calc\n      K ≤ k := le_max_left K k₁\n      _ ≤ 3 ^ k := by apply le_of_lt (Nat.lt_pow_self (by norm_num) k)\n  have h₆ : min_dist (r ^ 3 ^ k) > Real.exp (-γ * 3 ^ k) := by\n    have : Nat.pow 3 k = (3 : ℝ) ^ k := by simp -- 抽象化\n    calc\n      min_dist (r ^ 3 ^ k) = min_dist (r ^ (3 ^ k : ℝ)) := by rw [← Real.rpow_natCast, ← Real.rpow_natCast]; simp\n      _ > Real.exp (-γ * 3 ^ k) := by rw [← this]; apply h₄.right (3 ^ k) h₅\n  have h₇ : min_dist (r ^ 3 ^ k) ≤ Real.exp (-γ * 3 ^ k) := by\n    calc\n      min_dist (r ^ 3 ^ k) ≤ |(r : ℝ) ^ 3 ^ k - Nat.floor ((r : ℝ) ^ 3 ^ k)| := by\n        apply min_dist_floor\n        rw [h, ← Real.rpow_natCast]\n        apply Real.rpow_nonneg (by linarith [Mills_gt_one])\n      _ = |A ^ 3 ^ k - Nat.floor (A ^ 3 ^ k)| := by rw [h]\n      _ = |A ^ (3 : ℝ) ^ k - Nat.floor (A ^ (3 : ℝ) ^ k)| := by\n        rw [← Real.rpow_natCast, ← Real.rpow_natCast]\n        have : (Nat.cast : ℕ → ℝ) (Nat.pow 3 k) = (3 : ℝ) ^ (k : ℝ) := by simp --抽象化\n        rw [← this]; simp\n      _ = |A ^ (3 : ℝ) ^ k - p' k| := by rw [p'eqp''', p''', ← Real.rpow_natCast]\n      _ ≤ Real.exp (-γ * 3 ^ k) := h₃ k (le_max_right K k₁)\n  linarith\n","line":700,"end_line":736,"proof_status":"statement_formalized"}
{"id":"thm:BHP","label":"BHP","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"BHP","doc_url":null,"latex_link":"sect0001.html#thm:BHP","lean_link":null,"lean_decl_name":"BHP","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Mills","source_path":"./Mills/Defs.lean","lean4_code":"theorem BHP' (x : ℕ+) (h : BHP_cons_pnat ≤ x) : ∃ p : ℕ+, x ≤ p ∧ p ≤ x + (pn_pow_nnr x θ) ∧ Nat.Prime p := by sorry\n\n","line":98,"end_line":99,"proof_status":"statement_formalized"}
{"id":"thm:Mahler","label":"Mahler","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Mahler","doc_url":null,"latex_link":"sect0001.html#thm:Mahler","lean_link":null,"lean_decl_name":"Mahler","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}