{"id":22390638,"url":"https://github.com/jjaassoonn/transcendental","last_synced_at":"2025-03-26T21:45:16.506Z","repository":{"id":101612876,"uuid":"234444826","full_name":"jjaassoonn/transcendental","owner":"jjaassoonn","description":null,"archived":false,"fork":false,"pushed_at":"2020-09-09T13:54:55.000Z","size":4946,"stargazers_count":6,"open_issues_count":0,"forks_count":1,"subscribers_count":2,"default_branch":"master","last_synced_at":"2025-02-01T03:42:23.985Z","etag":null,"topics":["leanprover"],"latest_commit_sha":null,"homepage":"https://jjaassoonn.github.io","language":"HTML","has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":null,"status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/jjaassoonn.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":null,"code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null}},"created_at":"2020-01-17T01:23:52.000Z","updated_at":"2021-07-12T13:28:22.000Z","dependencies_parsed_at":"2023-06-06T13:15:14.643Z","dependency_job_id":null,"html_url":"https://github.com/jjaassoonn/transcendental","commit_stats":null,"previous_names":[],"tags_count":0,"template":false,"template_full_name":null,"repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/jjaassoonn%2Ftranscendental","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/jjaassoonn%2Ftranscendental/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/jjaassoonn%2Ftranscendental/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/jjaassoonn%2Ftranscendental/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/jjaassoonn","download_url":"https://codeload.github.com/jjaassoonn/transcendental/tar.gz/refs/heads/master","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":245743341,"owners_count":20665090,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":["leanprover"],"created_at":"2024-12-05T03:41:35.699Z","updated_at":"2025-03-26T21:45:16.501Z","avatar_url":"https://github.com/jjaassoonn.png","language":"HTML","funding_links":[],"categories":[],"sub_categories":[],"readme":"\n# Transcendental Numbers\n\nThis project is to prove several theorems in transcendental number theory:\n\n1. [x] Countability argument: abstract existence of transcendental number;\n2. [x] Liouvielle theorem and hence ![liouville number](https://latex.codecogs.com/gif.latex?\\sum_{i=0}^\\infty\u0026space;\\frac{1}{10^{n!}}) is transcendental;\n3. [x] ![e](https://latex.codecogs.com/gif.latex?e) is transcendental;\n4. [ ] ![pi](https://latex.codecogs.com/gif.latex?\\pi) is transcendental.\n\n## Part 1, countability argument\n\nThe main theorem is in [algebraic_countable_over_Z.lean](https://github.com/jjaassoonn/transcendental/blob/master/src/algebraic_countable_over_Z.lean#L731)\n\n```lean\ntheorem transcendental_number_exists : ∃ x : real, ¬ (is_algebraic ℤ x)\n```\n\nThe other version is in [algebraic_countable_over_Q.lean](https://github.com/jjaassoonn/transcendental/blob/master/src/algebraic_countable_over_Q.lean#L897)\n\n```lean\ntheorem transcendental_number_exists : ∃ x : real, ¬ (is_algebraic ℚ x)\n```\n\n## Part 2, Liouville theorem and an explicit Liouville number\n\nDefinition of the explicit Liouville number is in [liouville_theorem.lean](https://github.com/jjaassoonn/transcendental/blob/master/src/liouville_theorem.lean#L1136)\n\n```lean\ndef α := ∑' n, ten_pow_n_fact_inverse n\n```\n\nThe main theorem is in [liouville_theorem.lean](https://github.com/jjaassoonn/transcendental/blob/master/src/liouville_theorem.lean#L863):\n\n```lean\ntheorem liouville_numbers_transcendental : ∀ x : real, liouville_number x -\u003e ¬(is_algebraic ℤ x)\n\ntheorem transcendental_α : transcendental α := liouville_numbers_transcendental α liouville_α\n```\n\n## Part 3, the transcendence of e\n\nWe defined e in [e_transcendental.lean](https://github.com/jjaassoonn/transcendental/blob/699e50a6d262ee73ab20bfa6362ed637d4e88c77/src/e_transcendental.lean#L15) as :\n\n``` lean\ndef e : ℝ := real.exp 1\n```\n\nThe main theorem is at [e_transcendental.lean](https://github.com/jjaassoonn/transcendental/blob/699e50a6d262ee73ab20bfa6362ed637d4e88c77/src/e_transcendental.lean#L1798):\n\n```lean\ntheorem e_transcendental : ¬ is_algebraic ℤ e :=\n```\n\nAlmost immediately, we can prove\n```lean\ntheorem e_irrational : irrational e\n\ntheorem e_pow_transcendental (n : ℕ) (hn : n ≥ 1) : transcendental (e^n)\ntheorem e_pow_n_irrational (n : ℕ) (hn : n ≥ 1) : irrational (e ^ n)\n```\n\nPlease see [this](https://jjaassoonn.github.io/e_transcendence_doc.html) for an explanation of the proof of transcendence of $e$ with reference to Lean code.\n\nI haven't finished documentation (not even close), but you can click around the proves I documented so far\nat [e_trans_helpers2.lean](https://jjaassoonn.github.io/transcendental/html/e_trans_helpers2.html) and[e_transcendental.lean](https://jjaassoonn.github.io/transcendental/html/e_transcendental.html).\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fjjaassoonn%2Ftranscendental","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fjjaassoonn%2Ftranscendental","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fjjaassoonn%2Ftranscendental/lists"}