├── blueprint ├── lean_decls ├── src │ ├── blueprint.sty │ ├── web.paux │ ├── latexmkrc │ ├── macros │ │ ├── web.tex │ │ ├── common.tex │ │ └── print.tex │ ├── plastex.cfg │ ├── extra_styles.css │ ├── web.tex │ ├── print.tex │ └── content.tex ├── print │ └── print.pdf └── web │ ├── js │ ├── expatlib.wasm │ ├── graphvizlib.wasm │ ├── js.cookie.min.js │ ├── plastex.js │ ├── showmore.js │ └── svgxuse.js │ ├── styles │ ├── blueprint.css │ ├── showmore.css │ ├── extra_styles.css │ ├── amsthm.css │ ├── dep_graph.css │ ├── theme-white.css │ ├── theme-green.css │ └── theme-blue.css │ ├── index.html │ ├── sect0001.html │ ├── dep_graph_document.html │ └── symbol-defs.svg ├── lean-toolchain ├── ABCTrueAlmostAlways.lean ├── home_page ├── assets │ └── css │ │ └── style.scss ├── 404.html ├── index.md ├── _include │ └── mathjax.html ├── Gemfile ├── _config.yml ├── _layouts │ └── default.html └── Gemfile.lock ├── lakefile.toml ├── README.md ├── .gitignore └── lake-manifest.json /blueprint/lean_decls: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /lean-toolchain: -------------------------------------------------------------------------------- 1 | leanprover/lean4:v4.20.0 2 | -------------------------------------------------------------------------------- /blueprint/src/blueprint.sty: -------------------------------------------------------------------------------- 1 | \DeclareOption*{} 2 | \ProcessOptions 3 | 4 | \newcommand{\graphcolor}[3]{} -------------------------------------------------------------------------------- /blueprint/src/web.paux: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/morph-labs/lean-abc-true-almost-always/HEAD/blueprint/src/web.paux -------------------------------------------------------------------------------- /blueprint/print/print.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/morph-labs/lean-abc-true-almost-always/HEAD/blueprint/print/print.pdf -------------------------------------------------------------------------------- /blueprint/web/js/expatlib.wasm: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/morph-labs/lean-abc-true-almost-always/HEAD/blueprint/web/js/expatlib.wasm -------------------------------------------------------------------------------- /blueprint/web/js/graphvizlib.wasm: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/morph-labs/lean-abc-true-almost-always/HEAD/blueprint/web/js/graphvizlib.wasm -------------------------------------------------------------------------------- /blueprint/web/styles/blueprint.css: -------------------------------------------------------------------------------- 1 | a.github_link { 2 | font-weight: normal; 3 | font-size: 90%; 4 | text-decoration: none; 5 | color: inherit; 6 | } 7 | 8 | -------------------------------------------------------------------------------- /ABCTrueAlmostAlways.lean: -------------------------------------------------------------------------------- 1 | -- This module serves as the root of the `ABCTrueAlmostAlways` library. 2 | -- Import modules here that should be built as part of the library. 3 | import ABCTrueAlmostAlways.ABCTrueAlmostAlways 4 | -------------------------------------------------------------------------------- /blueprint/src/latexmkrc: -------------------------------------------------------------------------------- 1 | # This file configures the latexmk command you can use to compile 2 | # the pdf version of the blueprint 3 | $pdf_mode = 1; 4 | $pdflatex = 'xelatex -synctex=1'; 5 | @default_files = ('print.tex'); -------------------------------------------------------------------------------- /blueprint/web/styles/showmore.css: -------------------------------------------------------------------------------- 1 | p.hidden { 2 | display: none; 3 | } 4 | 5 | svg.showmore { 6 | font-size: 150%; 7 | margin: auto; 8 | min-width: 2rem; 9 | text-decoration: none; 10 | text-shadow: 1px 2px 0 rgba(0, 0, 0, 0.8); 11 | } 12 | 13 | -------------------------------------------------------------------------------- /home_page/assets/css/style.scss: -------------------------------------------------------------------------------- 1 | --- 2 | --- 3 | 4 | /* In this file you can add css rules overriding rules defined by your Jekyll theme. */ 5 | 6 | /* Import the CSS from the specified theme */ 7 | @import "{{ site.theme }}"; 8 | 9 | /* Put your rules below this line */ -------------------------------------------------------------------------------- /blueprint/src/macros/web.tex: -------------------------------------------------------------------------------- 1 | % In this file you should put macros to be used only by 2 | % the web version. Of course they should have a corresponding 3 | % version in macros/print.tex. 4 | % Typically the printed version could have more fancy decorations. 5 | % This will probably be a very short file. -------------------------------------------------------------------------------- /blueprint/src/plastex.cfg: -------------------------------------------------------------------------------- 1 | [general] 2 | renderer=HTML5 3 | copy-theme-extras=yes 4 | plugins=plastexdepgraph plastexshowmore leanblueprint 5 | 6 | [document] 7 | toc-depth=3 8 | toc-non-files=True 9 | 10 | [files] 11 | directory=../web/ 12 | split-level= 0 13 | 14 | [html5] 15 | localtoc-level=0 16 | extra-css=extra_styles.css 17 | mathjax-dollars=False -------------------------------------------------------------------------------- /lakefile.toml: -------------------------------------------------------------------------------- 1 | name = "ABCTrueAlmostAlways" 2 | version = "0.1.0" 3 | keywords = ["math"] 4 | defaultTargets = ["ABCTrueAlmostAlways"] 5 | 6 | [leanOptions] 7 | pp.unicode.fun = true # pretty-prints `fun a ↦ b` 8 | autoImplicit = false 9 | 10 | [[require]] 11 | name = "mathlib" 12 | scope = "leanprover-community" 13 | rev = "v4.20.0" 14 | 15 | [[require]] 16 | name = "checkdecls" 17 | git = "https://github.com/PatrickMassot/checkdecls.git" 18 | 19 | [[lean_lib]] 20 | name = "ABCTrueAlmostAlways" 21 | -------------------------------------------------------------------------------- /home_page/404.html: -------------------------------------------------------------------------------- 1 | --- 2 | permalink: /404.html 3 | layout: default 4 | --- 5 | 6 | 19 | 20 |
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404

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Page not found :(

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The requested page could not be found.

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-------------------------------------------------------------------------------- /home_page/index.md: -------------------------------------------------------------------------------- 1 | --- 2 | # Feel free to add content and custom Front Matter to this file. 3 | # To modify the layout, see https://jekyllrb.com/docs/themes/#overriding-theme-defaults 4 | 5 | # layout: home 6 | usemathjax: true 7 | --- 8 | 9 | Useful links: 10 | 11 | * [Zulip chat for Lean](https://leanprover.zulipchat.com/) for coordination 12 | * [Blueprint]({{ site.url }}/blueprint/) 13 | * [Blueprint as pdf]({{ site.url }}/blueprint.pdf) 14 | * [Dependency graph]({{ site.url }}/blueprint/dep_graph_document.html) 15 | * [Doc pages for this repository]({{ site.url }}/docs/) -------------------------------------------------------------------------------- /blueprint/src/extra_styles.css: -------------------------------------------------------------------------------- 1 | /* This file contains CSS tweaks for this blueprint. 2 | * As an example, we included CSS rules that put 3 | * a vertical line on the left of theorem statements 4 | * and proofs. 5 | * */ 6 | 7 | div.theorem_thmcontent { 8 | border-left: .15rem solid black; 9 | } 10 | 11 | div.proposition_thmcontent { 12 | border-left: .15rem solid black; 13 | } 14 | 15 | div.lemma_thmcontent { 16 | border-left: .1rem solid black; 17 | } 18 | 19 | div.corollary_thmcontent { 20 | border-left: .1rem solid black; 21 | } 22 | 23 | div.proof_content { 24 | border-left: .08rem solid grey; 25 | } 26 | -------------------------------------------------------------------------------- /blueprint/web/styles/extra_styles.css: -------------------------------------------------------------------------------- 1 | /* This file contains CSS tweaks for this blueprint. 2 | * As an example, we included CSS rules that put 3 | * a vertical line on the left of theorem statements 4 | * and proofs. 5 | * */ 6 | 7 | div.theorem_thmcontent { 8 | border-left: .15rem solid black; 9 | } 10 | 11 | div.proposition_thmcontent { 12 | border-left: .15rem solid black; 13 | } 14 | 15 | div.lemma_thmcontent { 16 | border-left: .1rem solid black; 17 | } 18 | 19 | div.corollary_thmcontent { 20 | border-left: .1rem solid black; 21 | } 22 | 23 | div.proof_content { 24 | border-left: .08rem solid grey; 25 | } 26 | -------------------------------------------------------------------------------- /home_page/_include/mathjax.html: -------------------------------------------------------------------------------- 1 | 2 | {% if page.usemathjax %} 3 | 4 | 13 | 14 | 17 | {% endif %} -------------------------------------------------------------------------------- /blueprint/src/macros/common.tex: -------------------------------------------------------------------------------- 1 | % In this file you should put all LaTeX macros and settings to be used both by 2 | % the pdf version and the web version. 3 | % This should be most of your macros. 4 | 5 | % The theorem-like environments defined below are those that appear by default 6 | % in the dependency graph. See the README of leanblueprint if you need help to 7 | % customize this. 8 | % The configuration below use the theorem counter for all those environments 9 | % (this is what the [theorem] arguments mean) and never resets it. 10 | % If you want for instance to number them within chapters then you can add 11 | % [chapter] at the end of the next line. 12 | \newtheorem{theorem}{Theorem} 13 | \newtheorem{proposition}[theorem]{Proposition} 14 | \newtheorem{lemma}[theorem]{Lemma} 15 | \newtheorem{corollary}[theorem]{Corollary} 16 | 17 | \theoremstyle{definition} 18 | \newtheorem{definition}[theorem]{Definition} -------------------------------------------------------------------------------- /home_page/Gemfile: -------------------------------------------------------------------------------- 1 | source "https://rubygems.org" 2 | 3 | # To upgrade, run `bundle update github-pages`. 4 | gem "github-pages", group: :jekyll_plugins 5 | # If you have any plugins, put them here! 6 | group :jekyll_plugins do 7 | #gem "jekyll-feed", "~> 0.12" 8 | end 9 | 10 | # Windows and JRuby does not include zoneinfo files, so bundle the tzinfo-data gem 11 | # and associated library. 12 | platforms :mingw, :x64_mingw, :mswin, :jruby do 13 | gem "tzinfo", "~> 1.2" 14 | gem "tzinfo-data" 15 | end 16 | 17 | # Performance-booster for watching directories on Windows. 18 | gem "wdm", "~> 0.1.1", :platforms => [:mingw, :x64_mingw, :mswin] 19 | 20 | # Lock `http_parser.rb` gem to `v0.6.x` on JRuby builds since newer versions of the gem 21 | # do not have a Java counterpart. 22 | gem "http_parser.rb", "~> 0.6.0", :platforms => [:jruby] 23 | 24 | # Used for locally serving the website. 25 | gem "webrick", "~> 1.7" -------------------------------------------------------------------------------- /blueprint/src/macros/print.tex: -------------------------------------------------------------------------------- 1 | % In this file you should put macros to be used only by 2 | % the printed version. Of course they should have a corresponding 3 | % version in macros/web.tex. 4 | % Typically the printed version could have more fancy decorations. 5 | % This should be a very short file. 6 | % 7 | % This file starts with dummy macros that ensure the pdf 8 | % compiler will ignore macros provided by plasTeX that make 9 | % sense only for the web version, such as dependency graph 10 | % macros. 11 | 12 | 13 | % Dummy macros that make sense only for web version. 14 | \newcommand{\lean}[1]{} 15 | \newcommand{\discussion}[1]{} 16 | \newcommand{\leanok}{} 17 | \newcommand{\mathlibok}{} 18 | \newcommand{\notready}{} 19 | % Make sure that arguments of \uses and \proves are real labels, by using invisible refs: 20 | % latex prints a warning if the label is not defined, but nothing is shown in the pdf file. 21 | % It uses LaTeX3 programming, this is why we use the expl3 package. 22 | \ExplSyntaxOn 23 | \NewDocumentCommand{\uses}{m} 24 | {\clist_map_inline:nn{#1}{\vphantom{\ref{##1}}}% 25 | \ignorespaces} 26 | \NewDocumentCommand{\proves}{m} 27 | {\clist_map_inline:nn{#1}{\vphantom{\ref{##1}}}% 28 | \ignorespaces} 29 | \ExplSyntaxOff -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # *The abc conjecture almost always* — autoformalized 2 | 3 | This is a completely machine-generated formalization of the classical theorem of de Bruijn, which bounds the exceptional set in the abc conjecture. We follow the proof laid out in this [expository note](https://arxiv.org/pdf/2505.13991). 4 | 5 | All statements, proofs, and documentation were created by Trinity, an autoformalization system developed by Morph Labs as part of the [Verified Superintelligence project](https://www.morph.so/blog/verified-superintelligence). 6 | 7 | The celebrated *abc* conjecture asserts that, for any `ε > 0`, all solutions to `a + b = c` in triples `a, b, c` of coprime integers must satisfy `rad(abc) ⩾ c ^ (1−ε)`, with finitely many exceptions. We obtain the first formalized theorem towards the *abc* conjecture, showing it is true almost always. Specifically, we formally verify that there are `O(N^{2/3})` such triples `(a, b, c) ∈ [1, N]^3`, which satisfy `rad(abc) < c ^ (1−ε)`. 8 | 9 | The final Lean output is in [ABCTrueAlmostAlways.lean](./ABCTrueAlmostAlways/ABCTrueAlmostAlways.lean). The human-supplied blueprint is in [content.tex](./blueprint/src/content.tex). 10 | 11 | ![Screenshot 2025-06-14 100620](https://github.com/user-attachments/assets/7b4cdf23-04ee-4cef-a78e-cb7c09f8a535) 12 | -------------------------------------------------------------------------------- /blueprint/web/styles/amsthm.css: -------------------------------------------------------------------------------- 1 | 2 | span[class$='_thmlabel'] 3 | { 4 | margin-left: .5rem; 5 | } 6 | 7 | div.theorem-style-plain div[class$='_thmheading'] { 8 | font-style: normal; 9 | font-weight: bold; 10 | 11 | } 12 | div.theorem-style-plain span[class$='_thmlabel']::after 13 | { 14 | content: '.'; 15 | } 16 | div.theorem-style-plain div[class$='_thmcontent'] { 17 | font-style: italic; 18 | font-weight: normal; 19 | 20 | } 21 | 22 | div.theorem-style-definition div[class$='_thmheading'] { 23 | font-style: normal; 24 | font-weight: bold; 25 | 26 | } 27 | div.theorem-style-definition span[class$='_thmlabel']::after 28 | { 29 | content: '.'; 30 | } 31 | div.theorem-style-remark div[class$='_thmheading'] { 32 | font-style: normal; 33 | font-weight: bold; 34 | 35 | } 36 | div.theorem-style-remark span[class$='_thmlabel']::after 37 | { 38 | content: '.'; 39 | } -------------------------------------------------------------------------------- /blueprint/src/web.tex: -------------------------------------------------------------------------------- 1 | % This file makes a web version of the blueprint 2 | % It should include all the \usepackage needed for this version. 3 | % The template includes standard AMS packages. 4 | % It is otherwise a very minimal preamble (you should probably at least 5 | % add cleveref and tikz-cd). 6 | 7 | \documentclass{report} 8 | 9 | \usepackage{amssymb, amsthm, amsmath} 10 | \usepackage{hyperref} 11 | \usepackage[showmore, dep_graph]{blueprint} 12 | % Blackboard Bold letters 13 | \newcommand{\C}{{\mathbb C}} 14 | \newcommand{\R}{{\mathbb R}} 15 | \newcommand{\Q}{{\mathbb Q}} 16 | \newcommand{\Z}{{\mathbb Z}} 17 | \newcommand{\N}{{\mathbb N}} 18 | \newcommand{\F}{{\mathbb F}} 19 | \newcommand{\E}{{\mathbb E}} 20 | \newcommand{\T}{{\mathbb T}} 21 | \newcommand{\D}{{\mathbb D}} 22 | 23 | % Math operators (the correct way to define them) 24 | \DeclareMathOperator\rad{rad} 25 | \DeclareMathOperator{\diag}{diag} 26 | 27 | % Other shortcuts 28 | \newcommand{\1}{{\mathbf 1}} 29 | \newcommand{\eps}{\varepsilon} % Using varepsilon (ε) which is often preferred over epsilon (ϵ) 30 | \renewcommand{\phi}{\varphi} % Personal preference for phi symbol 31 | 32 | 33 | \input{macros/common} 34 | \input{macros/web} 35 | 36 | \home{https://morph-labs.github.io/lean-abc-true-almost-always} 37 | \github{https://github.com/morph-labs/lean-abc-true-almost-always} 38 | \dochome{https://morph-labs.github.io/lean-abc-true-almost-always/docs} 39 | 40 | \title{abc exceptions} 41 | \author{morph} 42 | 43 | \begin{document} 44 | \maketitle 45 | \input{content} 46 | \end{document} 47 | -------------------------------------------------------------------------------- /home_page/_config.yml: -------------------------------------------------------------------------------- 1 | # Welcome to Jekyll! 2 | # 3 | # This config file is meant for settings that affect your whole blog, values 4 | # which you are expected to set up once and rarely edit after that. If you find 5 | # yourself editing this file very often, consider using Jekyll's data files 6 | # feature for the data you need to update frequently. 7 | # 8 | # For technical reasons, this file is *NOT* reloaded automatically when you use 9 | # 'bundle exec jekyll serve'. If you change this file, please restart the server process. 10 | # 11 | # If you need help with YAML syntax, here are some quick references for you: 12 | # https://learn-the-web.algonquindesign.ca/topics/markdown-yaml-cheat-sheet/#yaml 13 | # https://learnxinyminutes.com/docs/yaml/ 14 | # 15 | # Site settings 16 | # These are used to personalize your new site. If you look in the HTML files, 17 | # you will see them accessed via {{ site.title }}, {{ site.email }}, and so on. 18 | # You can create any custom variable you would like, and they will be accessible 19 | # in the templates via {{ site.myvariable }}. 20 | 21 | title: abc 22 | #email: your-email@example.com 23 | description: by greg-const 24 | baseurl: "" # the subpath of your site, e.g. /blog 25 | url: "https://morph-labs.github.io/lean-abc-true-almost-always" # the base hostname & protocol for your site, e.g. http://example.com 26 | twitter_username: 27 | github_username: morph-labs 28 | repository: morph-labs/lean-abc-true-almost-always 29 | 30 | # Build settings 31 | remote_theme: pages-themes/cayman@v0.2.0 32 | plugins: 33 | - jekyll-remote-theme 34 | - jekyll-github-metadata -------------------------------------------------------------------------------- /blueprint/web/js/js.cookie.min.js: -------------------------------------------------------------------------------- 1 | /*! js-cookie v3.0.1 | MIT */ 2 | !function(e,t){"object"==typeof exports&&"undefined"!=typeof module?module.exports=t():"function"==typeof define&&define.amd?define(t):(e=e||self,function(){var n=e.Cookies,o=e.Cookies=t();o.noConflict=function(){return e.Cookies=n,o}}())}(this,(function(){"use strict";function e(e){for(var t=1;t 2 | 3 | 4 | 5 | 6 | 7 | {% seo %} 8 | 9 | 11 | 12 | 13 | 14 | {% if jekyll.environment == "production" %} 15 | 17 | {% else %} 18 | 19 | {% endif %} 20 | {% include head-custom.html %} 21 | 22 | 23 | 24 | 25 | 26 | 37 | 38 |
39 | {{ content }} 40 | 41 |
42 | {% if site.github.is_project_page %} 43 | {{ site.github.repository_name }} 44 | is maintained by greg-const. Visit the GitHub repository for more information. 45 | {% endif %} 46 |
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48 | 49 | 50 | -------------------------------------------------------------------------------- /blueprint/web/js/plastex.js: -------------------------------------------------------------------------------- 1 | $(document).ready(function() { 2 | var icon = function($icon, $class, $id) { 3 | if ($id) { 4 | $id = ' id="'+$id+'"'; 5 | } else { 6 | $id = ''; 7 | } 8 | 9 | return '' 10 | }; 11 | 12 | $("#toc-toggle").click(function() { 13 | $("nav.toc").toggle() 14 | }); 15 | 16 | $("nav.toc").on("click", "span.expand-toc", 17 | function() { 18 | $(this).siblings("ul").slideToggle('fast'); 19 | 20 | if ($(this).html() == "▼") { 21 | $(this).html("▶"); 22 | } else { 23 | $(this).html("▼"); 24 | }; 25 | 26 | }) 27 | 28 | 29 | $("div.proof_content p:last").append('□') 30 | 31 | $("div.proof_heading").click( 32 | function() { 33 | var expand_span = $(this).children('span.expand-proof'); 34 | if ($(expand_span).html() == "▼") { 35 | $(expand_span).html("▶"); 36 | } else { 37 | $(expand_span).html("▼"); 38 | }; 39 | 40 | $(this).siblings("div.proof_content").slideToggle() 41 | }) 42 | 43 | $("a.proof").click( 44 | function() { 45 | var ref= $(this).attr('href').split('#')[1]; 46 | var proof = $('#'+ref) 47 | proof.show() 48 | proof.children('.proof_content').each( 49 | function() { 50 | var proof_content = $(this) 51 | proof_content.show().addClass('hilite') 52 | setTimeout(function(){ 53 | proof_content.removeClass('hilite') 54 | }, 1000); 55 | }) 56 | var expand_icon = proof.find('svg.expand-proof'); 57 | expand_icon.replaceWith(icon('cross', 'expand-proof')); 58 | }) 59 | 60 | 61 | $("button.modal").click( 62 | function() { 63 | $(this).next("div.modal-container").css('display', 'flex'); 64 | }) 65 | $("button.closebtn").click( 66 | function() { 67 | $(this).parent().parent().parent().hide(); 68 | }) 69 | }); 70 | -------------------------------------------------------------------------------- /blueprint/web/js/showmore.js: -------------------------------------------------------------------------------- 1 | $(document).ready(function() { 2 | showmore_update = function(showmore_level) { 3 | Cookies.set('showmore_level', showmore_level, { expires : 10, SameSite: "Lax" }); 4 | console.log('update', showmore_level); 5 | switch(showmore_level) { 6 | case 0: 7 | $("svg#showmore-minus").hide(); 8 | $("figure").hide(); 9 | $("svg#showmore-plus").show(); 10 | $("div.main-text > p").each( 11 | function(){ 12 | $(this).hide(); 13 | }); 14 | $("div.proof_wrapper").each( 15 | function(){ 16 | $(this).hide(); 17 | }); 18 | $("footer").hide(); 19 | break; 20 | case 1: 21 | $("svg#showmore-minus").show(); 22 | $("figure").show(); 23 | $("div.proof_wrapper").each( 24 | function(){ 25 | $(this).show(); 26 | }); 27 | $("svg#showmore-plus").show(); 28 | $("div.main-text > p").each( 29 | function(){ 30 | $(this).show(); 31 | }); 32 | $("div.proof_content").each( 33 | function(){ 34 | $(this).hide(); 35 | }); 36 | $("span.expand-proof").html("▶"); 37 | $("footer").show(); 38 | break; 39 | case 2: 40 | $("svg#showmore-minus").show(); 41 | $("svg#showmore-plus").hide(); 42 | $("div.main-text > p").each( 43 | function(){ 44 | $(this).show(); 45 | }); 46 | $("div.proof_wrapper").each( 47 | function(){ 48 | $(this).show(); 49 | }); 50 | $("div.proof_content").each( 51 | function(){ 52 | $(this).show(); 53 | }); 54 | $("span.expand-proof").html("▼"); 55 | } 56 | }; 57 | 58 | cookie_level = function(){ 59 | var showmore_level = parseInt(Cookies.get('showmore_level')); 60 | 61 | if (isNaN(showmore_level)) { 62 | return 1; 63 | } else { 64 | return showmore_level; 65 | } 66 | }; 67 | showmore_update(cookie_level()); 68 | 69 | $("svg#showmore-minus").click( 70 | function() { 71 | var showmore_level = cookie_level(); 72 | console.log('click ', showmore_level); 73 | if (showmore_level > 0) { 74 | showmore_level -= 1; 75 | showmore_update(showmore_level); 76 | } 77 | }) 78 | 79 | $("svg#showmore-plus").click( 80 | function() { 81 | var showmore_level = cookie_level(); 82 | console.log('click ', showmore_level); 83 | if (showmore_level < 2) { 84 | showmore_level += 1; 85 | showmore_update(showmore_level); 86 | } 87 | }) 88 | }) 89 | -------------------------------------------------------------------------------- /blueprint/web/index.html: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 10 | 12 | 13 | 14 | 15 | abc 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
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76 | 77 | 85 | 86 | 87 | 88 | 89 | 90 | 91 | 92 | -------------------------------------------------------------------------------- /blueprint/src/print.tex: -------------------------------------------------------------------------------- 1 | %====================================================================== 2 | % PREAMBLE 3 | %====================================================================== 4 | 5 | \documentclass[a4paper]{report} 6 | 7 | % --- Page Layout --- 8 | \usepackage{geometry} % Modern package for page margins. Replaces fullpage. 9 | % \geometry{margin=1in} % Example: uncomment and adjust to set margins. 10 | 11 | % --- Core LaTeX & Math Packages --- 12 | \usepackage{expl3} % Required by leanblueprint for dependency graph logic. 13 | \usepackage{mathtools} % Loads amsmath, provides many improvements. 14 | \usepackage{amsthm} % For theorem environments (but leanblueprint may override). 15 | \usepackage{amssymb} % For symbols like \mathbb, though unicode-math also provides them. 16 | 17 | % --- Hyperlinks and Lists --- 18 | \usepackage{enumitem} % For better control over lists. 19 | \usepackage[unicode,colorlinks=true,linkcolor=blue,urlcolor=magenta,citecolor=blue]{hyperref} 20 | 21 | % --- Font and Encoding (for XeLaTeX/LuaLaTeX) --- 22 | % This is a major package. It MUST come after math packages like amsmath. 23 | % It provides modern font support and makes amssymb/amsfonts largely redundant. 24 | \usepackage[warnings-off={mathtools-colon,mathtools-overbracket}]{unicode-math} 25 | 26 | % --- Custom Commands --- 27 | % Blackboard Bold letters 28 | \newcommand{\C}{{\mathbb C}} 29 | \newcommand{\R}{{\mathbb R}} 30 | \newcommand{\Q}{{\mathbb Q}} 31 | \newcommand{\Z}{{\mathbb Z}} 32 | \newcommand{\N}{{\mathbb N}} 33 | \newcommand{\F}{{\mathbb F}} 34 | \newcommand{\E}{{\mathbb E}} 35 | \newcommand{\T}{{\mathbb T}} 36 | \newcommand{\D}{{\mathbb D}} 37 | 38 | % Math operators (the correct way to define them) 39 | \DeclareMathOperator\rad{rad} 40 | \DeclareMathOperator{\diag}{diag} 41 | 42 | % Other shortcuts 43 | \newcommand{\1}{{\mathbf 1}} 44 | \newcommand{\eps}{\varepsilon} % Using varepsilon (ε) which is often preferred over epsilon (ϵ) 45 | \renewcommand{\phi}{\varphi} % Personal preference for phi symbol 46 | 47 | % For Lie algebras, use \mathfrak (modern) instead of \frak (old) 48 | \newcommand{\gl}{\mathfrak{gl}} 49 | \newcommand{\so}{\mathfrak{so}} 50 | % Do NOT define \sl, it's a built-in LaTeX command. 51 | 52 | % --- LeanBlueprint Macros --- 53 | % These files from leanblueprint will set up theorem environments, etc. 54 | % It's important they come AFTER your packages and custom commands. 55 | \input{macros/common} 56 | \input{macros/print} 57 | 58 | %====================================================================== 59 | % DOCUMENT 60 | %====================================================================== 61 | 62 | \title{abc} 63 | \author{morph} 64 | 65 | \begin{document} 66 | \maketitle 67 | % The content file contains all the chapters and sections. 68 | \input{content} 69 | \end{document} 70 | -------------------------------------------------------------------------------- /blueprint/web/styles/dep_graph.css: -------------------------------------------------------------------------------- 1 | header { 2 | height: 3rem; 3 | justify-content: center; 4 | display: flex; 5 | } 6 | 7 | header a { 8 | margin-right: auto; 9 | text-decoration: none; 10 | color: inherit; 11 | } 12 | 13 | header a:visited { 14 | color: inherit; 15 | } 16 | 17 | header h1 { 18 | position: absolute; 19 | } 20 | 21 | div#graph { 22 | width: 100%; 23 | height: 90vh; 24 | resize: both; 25 | overflow: hidden; } 26 | 27 | 28 | div#statements { 29 | display: none; 30 | position: absolute; 31 | bottom: 0; 32 | width: 100%; 33 | background-color: #e8e8e8; /* FIXME: use sass */ 34 | } 35 | 36 | #Legend span.title { 37 | font-size: 150%; 38 | font-weight: bold; 39 | display: flex; 40 | margin-right: 2rem; 41 | cursor: pointer; 42 | } 43 | 44 | #Legend dl { 45 | position: absolute; 46 | top: 6rem; 47 | left:1.5rem; 48 | font-size: 120%; 49 | display: none; 50 | max-width: 30%; 51 | background: white 52 | } 53 | 54 | #Legend div.btn { 55 | margin-left: .5rem; 56 | margin-top: .1rem; 57 | } 58 | 59 | #Legend dt { 60 | font-weight: bold; 61 | } 62 | 63 | #Legend dt::after { 64 | content: ":"; 65 | } 66 | 67 | #Legend dd { 68 | margin-left: .5rem; 69 | margin-right: 1rem; 70 | display: inline; 71 | } 72 | 73 | 74 | div.bar { 75 | display: block; 76 | width: 22px; 77 | height: 3px; 78 | border-radius: 1px; 79 | margin-top: 2px; 80 | margin-bot: 2px; 81 | background: black; 82 | } 83 | 84 | 85 | button.dep-closebtn 86 | { 87 | position: absolute; 88 | top: 1rem; 89 | right: 1rem; 90 | font-size: 100%; 91 | font-weight: bold; 92 | 93 | background: Transparent; 94 | border: none; 95 | 96 | text-decoration: none; 97 | 98 | /* color: #fff;*/ 99 | cursor: pointer; 100 | } 101 | 102 | div.dep-modal-container { 103 | position: fixed; 104 | z-index: $modal-z-index; 105 | top: 0; 106 | left: 0; 107 | 108 | display: none; 109 | 110 | width: 90vw; 111 | margin-top: 4rem; 112 | margin-left: 5vw; 113 | margin-right: 5vw; 114 | } 115 | div.dep-modal-content { 116 | font-weight: normal; 117 | 118 | overflow: auto; 119 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The \(abc\) conjecture is true almost always — Expanded statements. (v4)
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Notation:

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Here \(f(n) = O(g(n))\) means that \(|f(n)| \le C\, g(n)\) for all \(n{\gt}0\) where \(C{\gt}0\) is an absolute constant; while \(f(n) = O_{\epsilon }(g(n))\) means \(|f(n)| \le C\, g(n)\) for some \(C=C_{\epsilon }{\gt}0\) depending on \({\epsilon }\). \(f(n) = o(1)\) means that \(\lim _{n\to \infty }f(n) = 0\).

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() We have \(2^x \ge x+1\) for any real \(x\ge 1\).

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() We have \(2^x \ge x/2+1\) for any real \(x\ge 0\).

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() Every integer \(n\ge 1\) factors uniquely into a product of primes \(n = p_1^{a_1}\cdots p_k^{a_k}\).

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2.0) 192 | jekyll-theme-minimal (0.2.0) 193 | jekyll (> 3.5, < 5.0) 194 | jekyll-seo-tag (~> 2.0) 195 | jekyll-theme-modernist (0.2.0) 196 | jekyll (> 3.5, < 5.0) 197 | jekyll-seo-tag (~> 2.0) 198 | jekyll-theme-primer (0.6.0) 199 | jekyll (> 3.5, < 5.0) 200 | jekyll-github-metadata (~> 2.9) 201 | jekyll-seo-tag (~> 2.0) 202 | jekyll-theme-slate (0.2.0) 203 | jekyll (> 3.5, < 5.0) 204 | jekyll-seo-tag (~> 2.0) 205 | jekyll-theme-tactile (0.2.0) 206 | jekyll (> 3.5, < 5.0) 207 | jekyll-seo-tag (~> 2.0) 208 | jekyll-theme-time-machine (0.2.0) 209 | jekyll (> 3.5, < 5.0) 210 | jekyll-seo-tag (~> 2.0) 211 | jekyll-titles-from-headings (0.5.3) 212 | jekyll (>= 3.3, < 5.0) 213 | jekyll-watch (2.2.1) 214 | listen (~> 3.0) 215 | jemoji (0.12.0) 216 | gemoji (~> 3.0) 217 | html-pipeline (~> 2.2) 218 | jekyll (>= 3.0, < 5.0) 219 | kramdown (2.3.2) 220 | rexml 221 | kramdown-parser-gfm (1.1.0) 222 | kramdown (~> 2.0) 223 | liquid (4.0.3) 224 | listen (3.7.1) 225 | rb-fsevent (~> 0.10, >= 0.10.3) 226 | rb-inotify (~> 0.9, >= 0.9.10) 227 | mercenary (0.3.6) 228 | minima (2.5.1) 229 | jekyll (>= 3.5, < 5.0) 230 | jekyll-feed (~> 0.9) 231 | jekyll-seo-tag (~> 2.1) 232 | minitest (5.15.0) 233 | multipart-post (2.1.1) 234 | nokogiri (1.13.6-x86_64-linux) 235 | racc (~> 1.4) 236 | octokit (4.22.0) 237 | faraday (>= 0.9) 238 | sawyer (~> 0.8.0, >= 0.5.3) 239 | pathutil (0.16.2) 240 | forwardable-extended (~> 2.6) 241 | public_suffix (4.0.7) 242 | racc (1.6.0) 243 | rb-fsevent (0.11.1) 244 | rb-inotify (0.10.1) 245 | ffi (~> 1.0) 246 | rexml (3.2.5) 247 | rouge (3.26.0) 248 | ruby2_keywords (0.0.5) 249 | rubyzip (2.3.2) 250 | safe_yaml (1.0.5) 251 | sass (3.7.4) 252 | sass-listen (~> 4.0.0) 253 | sass-listen (4.0.0) 254 | rb-fsevent (~> 0.9, >= 0.9.4) 255 | rb-inotify (~> 0.9, >= 0.9.7) 256 | sawyer (0.8.2) 257 | addressable (>= 2.3.5) 258 | faraday (> 0.8, < 2.0) 259 | simpleidn (0.2.1) 260 | unf (~> 0.1.4) 261 | terminal-table (1.8.0) 262 | unicode-display_width (~> 1.1, >= 1.1.1) 263 | thread_safe (0.3.6) 264 | typhoeus (1.4.0) 265 | ethon (>= 0.9.0) 266 | tzinfo (1.2.9) 267 | thread_safe (~> 0.1) 268 | unf (0.1.4) 269 | unf_ext 270 | unf_ext (0.0.8.1) 271 | unicode-display_width (1.8.0) 272 | webrick (1.7.0) 273 | zeitwerk (2.5.4) 274 | 275 | PLATFORMS 276 | x86_64-linux 277 | 278 | DEPENDENCIES 279 | github-pages 280 | http_parser.rb (~> 0.6.0) 281 | tzinfo (~> 1.2) 282 | tzinfo-data 283 | wdm (~> 0.1.1) 284 | webrick (~> 1.7) 285 | 286 | BUNDLED WITH 287 | 2.3.14 -------------------------------------------------------------------------------- /blueprint/web/js/svgxuse.js: -------------------------------------------------------------------------------- 1 | /*! 2 | * @copyright Copyright (c) 2016 IcoMoon.io 3 | * @license Licensed under MIT license 4 | * See https://github.com/Keyamoon/svgxuse 5 | * @version 1.1.20 6 | */ 7 | /*jslint browser: true */ 8 | /*global XDomainRequest, MutationObserver, window */ 9 | (function () { 10 | 'use strict'; 11 | if (window && window.addEventListener) { 12 | var cache = Object.create(null); // holds xhr objects to prevent multiple requests 13 | var checkUseElems; 14 | var tid; // timeout id 15 | var debouncedCheck = function () { 16 | clearTimeout(tid); 17 | tid = setTimeout(checkUseElems, 100); 18 | }; 19 | var unobserveChanges = function () { 20 | return; 21 | }; 22 | var observeChanges = function () { 23 | var observer; 24 | window.addEventListener('resize', debouncedCheck, false); 25 | window.addEventListener('orientationchange', debouncedCheck, false); 26 | if (window.MutationObserver) { 27 | observer = new MutationObserver(debouncedCheck); 28 | observer.observe(document.documentElement, { 29 | childList: true, 30 | subtree: true, 31 | attributes: true 32 | }); 33 | unobserveChanges = function () { 34 | try { 35 | observer.disconnect(); 36 | window.removeEventListener('resize', debouncedCheck, false); 37 | window.removeEventListener('orientationchange', debouncedCheck, false); 38 | } catch (ignore) {} 39 | }; 40 | } else { 41 | document.documentElement.addEventListener('DOMSubtreeModified', debouncedCheck, false); 42 | unobserveChanges = function () { 43 | document.documentElement.removeEventListener('DOMSubtreeModified', debouncedCheck, false); 44 | window.removeEventListener('resize', debouncedCheck, false); 45 | window.removeEventListener('orientationchange', debouncedCheck, false); 46 | }; 47 | } 48 | }; 49 | var createRequest = function (url) { 50 | // In IE 9, cross domain requests can only be sent using XDomainRequest. 51 | // XDomainRequest would fail if CORS headers are not set. 52 | // Therefore, XDomainRequest should only be used with cross domain requests. 53 | function getHostname(href) { 54 | var a = document.createElement('a'); 55 | a.href = href; 56 | return a.hostname; 57 | } 58 | var Request; 59 | var hname = location.hostname; 60 | var hname2; 61 | if (window.XMLHttpRequest) { 62 | Request = new XMLHttpRequest(); 63 | hname2 = getHostname(url); 64 | if (Request.withCredentials === undefined && hname2 !== '' && hname2 !== hname) { 65 | Request = XDomainRequest || undefined; 66 | } else { 67 | Request = XMLHttpRequest; 68 | } 69 | } 70 | return Request; 71 | }; 72 | var xlinkNS = 'http://www.w3.org/1999/xlink'; 73 | checkUseElems = function () { 74 | var base; 75 | var bcr; 76 | var fallback = ''; // optional fallback URL in case no base path to SVG file was given and no symbol definition was found. 77 | var hash; 78 | var i; 79 | var inProgressCount = 0; 80 | var isHidden; 81 | var Request; 82 | var url; 83 | var uses; 84 | var xhr; 85 | function observeIfDone() { 86 | // If done with making changes, start watching for chagnes in DOM again 87 | inProgressCount -= 1; 88 | if (inProgressCount === 0) { // if all xhrs were resolved 89 | unobserveChanges(); // make sure to remove old handlers 90 | observeChanges(); // watch for changes to DOM 91 | } 92 | } 93 | function attrUpdateFunc(spec) { 94 | return function () { 95 | if (cache[spec.base] !== true) { 96 | spec.useEl.setAttributeNS(xlinkNS, 'xlink:href', '#' + spec.hash); 97 | } 98 | }; 99 | } 100 | function onloadFunc(xhr) { 101 | return function () { 102 | var body = document.body; 103 | var x = document.createElement('x'); 104 | var svg; 105 | xhr.onload = null; 106 | x.innerHTML = xhr.responseText; 107 | svg = x.getElementsByTagName('svg')[0]; 108 | if (svg) { 109 | svg.setAttribute('aria-hidden', 'true'); 110 | svg.style.position = 'absolute'; 111 | svg.style.width = 0; 112 | svg.style.height = 0; 113 | svg.style.overflow = 'hidden'; 114 | body.insertBefore(svg, body.firstChild); 115 | } 116 | observeIfDone(); 117 | }; 118 | } 119 | function onErrorTimeout(xhr) { 120 | return function () { 121 | xhr.onerror = null; 122 | xhr.ontimeout = null; 123 | observeIfDone(); 124 | }; 125 | } 126 | unobserveChanges(); // stop watching for changes to DOM 127 | // find all use elements 128 | uses = document.getElementsByTagName('use'); 129 | for (i = 0; i < uses.length; i += 1) { 130 | try { 131 | bcr = uses[i].getBoundingClientRect(); 132 | } catch (ignore) { 133 | // failed to get bounding rectangle of the use element 134 | bcr = false; 135 | } 136 | url = uses[i].getAttributeNS(xlinkNS, 'href').split('#'); 137 | base = url[0]; 138 | hash = url[1]; 139 | isHidden = bcr && bcr.left === 0 && bcr.right === 0 && bcr.top === 0 && bcr.bottom === 0; 140 | if (bcr && bcr.width === 0 && bcr.height === 0 && !isHidden) { 141 | // the use element is empty 142 | // if there is a reference to an external SVG, try to fetch it 143 | // use the optional fallback URL if there is no reference to an external SVG 144 | if (fallback && !base.length && hash && !document.getElementById(hash)) { 145 | base = fallback; 146 | } 147 | if (base.length) { 148 | // schedule updating xlink:href 149 | xhr = cache[base]; 150 | if (xhr !== true) { 151 | // true signifies that prepending the SVG was not required 152 | setTimeout(attrUpdateFunc({ 153 | useEl: uses[i], 154 | base: base, 155 | hash: hash 156 | }), 0); 157 | } 158 | if (xhr === undefined) { 159 | Request = createRequest(base); 160 | if (Request !== undefined) { 161 | xhr = new Request(); 162 | cache[base] = xhr; 163 | xhr.onload = onloadFunc(xhr); 164 | xhr.onerror = onErrorTimeout(xhr); 165 | xhr.ontimeout = onErrorTimeout(xhr); 166 | xhr.open('GET', base); 167 | xhr.send(); 168 | inProgressCount += 1; 169 | } 170 | } 171 | } 172 | } else { 173 | if (!isHidden) { 174 | if (cache[base] === undefined) { 175 | // remember this URL if the use element was not empty and no request was sent 176 | cache[base] = true; 177 | } else if (cache[base].onload) { 178 | // if it turns out that prepending the SVG is not necessary, 179 | // abort the in-progress xhr. 180 | cache[base].abort(); 181 | delete cache[base].onload; 182 | cache[base] = true; 183 | } 184 | } else if (base.length && cache[base]) { 185 | attrUpdateFunc({ 186 | useEl: uses[i], 187 | base: base, 188 | hash: hash 189 | })(); 190 | } 191 | } 192 | } 193 | uses = ''; 194 | inProgressCount += 1; 195 | observeIfDone(); 196 | }; 197 | // The load event fires when all resources have finished loading, which allows detecting whether SVG use elements are empty. 198 | window.addEventListener('load', function winLoad() { 199 | window.removeEventListener('load', winLoad, false); // to prevent memory leaks 200 | tid = setTimeout(checkUseElems, 0); 201 | }, false); 202 | } 203 | }()); 204 | -------------------------------------------------------------------------------- /blueprint/web/styles/theme-white.css: -------------------------------------------------------------------------------- 1 | @charset 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a{padding-left:4rem}ul.sub-toc-1,ul.sub-toc-2,ul.sub-toc-3,ul.sub-toc-4{display:none}span.expand-toc{min-width:.7rem;width:.8rem;height:.8rem;padding:0;padding-right:.5rem;font-size:125%}span.expand-toc,svg.close-toc{cursor:pointer;text-align:center;color:#fff;background-color:Transparent}svg.close-toc{min-width:1.3rem;min-height:1.3rem;margin:.5rem;margin-left:auto;display:none}nav.active svg.close-toc{display:inline-block}ul.active{display:block}code.verb{font-family:monospace;font-style:normal;font-weight:400}pre.verbatim{margin:1rem 2rem;background-color:#dbdbdb;padding:.5rem} -------------------------------------------------------------------------------- /blueprint/web/symbol-defs.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | mindmap 5 | 6 | 7 | 8 | pencil 9 | 10 | 11 | 12 | lifebuoy 13 | 14 | 15 | 16 | cogs 17 | 18 | 19 | 20 | list-numbered 21 | 22 | 23 | 24 | list2 25 | 26 | 27 | 28 | menu 29 | 30 | 31 | 32 | eye-plus 33 | 34 | 35 | 36 | 37 | eye-minus 38 | 39 | 40 | 41 | 42 | smile 43 | 44 | 45 | 46 | sad 47 | 48 | 49 | 50 | plus 51 | 52 | 53 | 54 | cross 55 | 56 | 57 | 58 | arrow-up 59 | 60 | 61 | 62 | arrow-right 63 | 64 | 65 | 66 | arrow-left 67 | 68 | 69 | 70 | warning 71 | 72 | 73 | 74 | 75 | 76 | question 77 | 78 | 79 | 80 | 81 | -------------------------------------------------------------------------------- /blueprint/src/content.tex: -------------------------------------------------------------------------------- 1 | \chapter{smallpaper} 2 | \begin{lemma}\label{lem1} \lean{two_rpow_ge_add_one}\leanok 3 | We have $2^x \ge x+1$ for any real $x\ge1$. 4 | \end{lemma} 5 | \begin{proof} 6 | \leanok 7 | \end{proof} 8 | 9 | \begin{lemma}\label{lem2} \lean{two_rpow_ge_half_add_one} 10 | We have $2^x \ge x/2+1$ for any real $x\ge0$. 11 | \end{lemma} 12 | \begin{proof} 13 | \leanok 14 | \end{proof} 15 | 16 | 17 | \begin{theorem}[Fundamental theorem of arithmetic] \label{thm3} \lean{fundamental_theorem_of_arithmetic}\leanok 18 | Every integer $n\ge1$ factors uniquely into a product of primes $n = p_1^{a_1}\cdots p_k^{a_k}$. 19 | \end{theorem} 20 | \begin{proof} 21 | \leanok 22 | \end{proof} 23 | 24 | \begin{definition}[Divisor function] \label{def4} \lean{tau}\leanok 25 | $\tau(n)$ denotes the number of divisors of an integer $n\ge1$. 26 | \end{definition} 27 | 28 | \begin{lemma} \label{lem5} \lean{tau_eq_prod_factorization_add_one}\leanok 29 | If $n = p_1^{a_1}\cdots p_k^{a_k}$ is the prime factorization of an integer, then the divisor function evaluated at $n$ equals $\tau(n) = (a_1+1)\cdots(a_k+1)$. 30 | \end{lemma} 31 | \begin{proof} 32 | \leanok 33 | \end{proof} 34 | 35 | \begin{lemma} \label{lem6} \lean{tau_n_div_n_rpow_eps_eq_prod} 36 | If $n = p_1^{a_1}\cdots p_k^{a_k}$ is the prime factorization of an integer, then 37 | \begin{equation}\label{eq:taunn} 38 | \frac{\tau(n)}{n^\eps} = \prod_{i\le k}\frac{a_i+1}{p_i^{a_i \eps}}. 39 | \end{equation} 40 | \end{lemma} 41 | \begin{proof}\leanok 42 | \uses{def4, lem5} 43 | Use definition \ref{def4}, lemma \ref{lem5}. 44 | \end{proof} 45 | 46 | 47 | \begin{lemma} \label{lem7} \lean{lemma7} 48 | Let $p_i,a_i\ge1$ and $\eps>0$. If $p_i^{\eps}\ge 2$ then $\frac{a_i+1}{p_i^{a_i \eps}}\le \frac{a_i+1}{2^{a_i}}\le 1$. 49 | \end{lemma} 50 | \begin{proof}\leanok 51 | \uses{lem1} 52 | Use lemma \ref{lem1} with $x=a_i$. 53 | \end{proof} 54 | 55 | \begin{lemma} \label{lem8} \lean{lemma8} 56 | Let $p_i,a_i\ge1$ and $\eps>0$. If $p_i^{\eps}< 2$ then $\frac{a_i+1}{p_i^{a_i \eps}}\le 2/\eps$. 57 | \end{lemma} 58 | \begin{proof}\leanok 59 | \uses{lem2} 60 | Use lemma \ref{lem2} with $x=\eps$. 61 | \end{proof} 62 | 63 | \begin{lemma} \label{lem9} \lean{lemma9} 64 | If $\eps>0$, $a_1,\ldots,a_k\ge1$ are integers, and $p_1,\ldots, p_k$ are primes, then 65 | \[ 66 | \prod_{i\le k}\frac{a_i+1}{p_i^{a_i \eps}} = \prod_{i\le k, \, p_i^{\eps}\ge 2}\frac{a_i+1}{p_i^{a_i \eps}}\prod_{i\le k, \, p_i^{\eps}< 2}\frac{a_i+1}{p_i^{a_i \eps}} 67 | \] 68 | \end{lemma} 69 | \begin{proof}\leanok 70 | \uses{} 71 | Uses no previous result 72 | \end{proof} 73 | 74 | \begin{lemma} \label{lem10} \lean{lemma10} 75 | If $\eps>0$, $a_1,\ldots,a_k\ge1$ are integers, and $p_1,\ldots, p_k$ are primes, then 76 | \[ 77 | \prod_{i\le k, \, p_i^{\eps}\ge 2}\frac{a_i+1}{p_i^{a_i \eps}} \le 1 78 | \] 79 | \end{lemma} 80 | \begin{proof}\leanok 81 | \uses{lem7} 82 | Uses lemma \ref{lem7} 83 | \end{proof} 84 | 85 | \begin{lemma} \label{lem11} \lean{lemma11} 86 | If $\eps>0$, $a_1,\ldots,a_k\ge1$ are integers, and $p_1,\ldots, p_k$ are primes, then 87 | \[ 88 | \prod_{i\le k, \, p_i^{\eps}< 2}\frac{a_i+1}{p_i^{a_i \eps}} \le \prod_{i\le k, \, p_i^{\eps}< 2}\frac{2}{\eps} 89 | \] 90 | \end{lemma} 91 | \begin{proof}\leanok 92 | \uses{lem8} 93 | Uses lemma \ref{lem8} 94 | \end{proof} 95 | 96 | 97 | \begin{lemma} \label{lem12} \lean{lemma12} 98 | If $\eps>0$, $a_1,\ldots,a_k\ge1$ are integers, and $p_1,\ldots, p_k$ are primes, then 99 | \[ 100 | \prod_{i\le k, \, p_i^{\eps}< 2}\frac{2}{\eps} \le (2/\eps)^{2^{1/\eps}} 101 | \] 102 | \end{lemma} 103 | \begin{proof}\leanok 104 | \uses{} 105 | Uses no previous result 106 | \end{proof} 107 | 108 | 109 | \begin{lemma} \label{lem13} \lean{lemma13} 110 | If $\eps>0$, $a_1,\ldots,a_k\ge1$ are integers, and $p_1,\ldots, p_k$ are primes, then 111 | \[ 112 | \prod_{i\le k}\frac{a_i+1}{p_i^{a_i \eps}} \le (2/\eps)^{2^{1/\eps}} 113 | \] 114 | \end{lemma} 115 | \begin{proof}\leanok 116 | \uses{lem9, lem10, lem11, lem12} 117 | Uses lemmas \ref{lem9}, \ref{lem10}, \ref{lem11}, \ref{lem12} 118 | \end{proof} 119 | 120 | \begin{lemma} \label{lem14} \lean{lemma14} 121 | Let $\eps>0$. Then $(2/\eps)^{2^{1/\eps}} \le O_\eps(1)$. 122 | \end{lemma} 123 | \begin{proof}\leanok 124 | \uses{} 125 | Uses definition of $O_\eps(1)$. 126 | \end{proof} 127 | 128 | \begin{lemma} \label{lem15} \lean{lemma15} 129 | Let $\eps>0$ and $n\ge1$. Then $\frac{\tau(n)}{n^\eps}\le O_\eps(1)$ 130 | \end{lemma} 131 | \begin{proof}\leanok 132 | \uses{lem6, lem13, lem14} 133 | Uses lemma \ref{lem6}, \ref{lem13}, \ref{lem14} 134 | \end{proof} 135 | 136 | \begin{lemma} \label{lem16} \lean{lemma16} 137 | Let $\eps>0$ and $n\ge1$. Then $\tau(n) \le O_\eps(n^\eps)$. 138 | \end{lemma} 139 | \begin{proof}\leanok 140 | \uses{lem15} 141 | Uses lemma \ref{lem15} 142 | \end{proof} 143 | 144 | \begin{lemma} \label{lem17} \lean{lemma17} 145 | We have $\tau(n) \le O_\eps(n^\eps)$ for any integer $n\ge1$ and any $\eps>0$. 146 | \end{lemma} 147 | \begin{proof}\leanok 148 | \uses{lem16} 149 | Uses lemma \ref{lem16} 150 | \end{proof} 151 | 152 | \begin{theorem}[Divisor bound] \label{lem:div} \label{thm18} \lean{divisor_bound_tau_le_n_pow_o_one} 153 | We have $\tau(n) \le n^{o(1)}$ for any integer $n\ge1$. 154 | \end{theorem} 155 | \begin{proof}\leanok 156 | \uses{lem17} 157 | Uses lemma \ref{lem17} and definition of $o(1)$. 158 | \end{proof} 159 | 160 | 161 | \begin{definition}[Radical] \label{def19} \lean{rad}\leanok 162 | For an integer $n\ge1$ define the radical $\rad(n)=\prod_{p\mid n}p$. 163 | \end{definition} 164 | 165 | \begin{lemma} \label{lem20} \lean{rad_eq_prod_distinct_prime_factors} 166 | If $n = p_1^{a_1}\cdots p_k^{a_k}$ is the prime factorization of an integer, then $\rad(n)=p_1\cdots p_k$. 167 | \end{lemma} 168 | \begin{proof}\leanok 169 | \uses{def19} 170 | Uses definition \ref{def19}. 171 | \end{proof} 172 | 173 | \begin{lemma} \label{lem21} \lean{rad_mul_of_coprime} 174 | $\rad(n)$ is a multiplicative function. That is, if $a,b\ge1$ are integers with $\gcd(a,b)=1$ then $\rad(ab) = \rad(a)\rad(b)$. 175 | \end{lemma} 176 | \begin{proof}\leanok 177 | \uses{thm3, lem20} 178 | Uses lemma \ref{lem20} and theorem \ref{thm3}. 179 | \end{proof} 180 | 181 | \begin{lemma} \label{lem22} \lean{rad_abc_of_coprime} 182 | If $a,b\ge1$ are integers with $\gcd(a,bc)=1$ and $\gcd(b,c)=1$ then $\rad(abc) = \rad(a)\rad(b)\rad(c)$. 183 | \end{lemma} 184 | \begin{proof}\leanok 185 | \uses{lem21} 186 | Uses lemma \ref{lem21} with $\{a,bc\}$, and again uses lemma \ref{lem21} with $\{b,c\}$ 187 | \end{proof} 188 | 189 | \begin{lemma} \label{lem23} \lean{lemma23} 190 | If $a,b\ge1$ are integers with $\gcd(a,bc)=1$ and $\gcd(b,c)=1$ then $\rad(ab) \rad(ac)\rad(bc) = (\rad(abc))^2$. 191 | \end{lemma} 192 | \begin{proof}\leanok 193 | \uses{lem21, lem22} 194 | Uses lemma \ref{lem22}, and then uses lemma \ref{lem21} three times, with $\{a,b\}$ and $\{a,c\}$ and $\{b,c\}$. 195 | \end{proof} 196 | 197 | 198 | \begin{lemma} \label{lem24} \lean{lemma24} 199 | Let $p_1<\cdots0$ and $1\le n\le N$, then $\frac{1}{n^{\eps}} \ge \frac{1}{N^{\eps}}$. 231 | \end{lemma} 232 | \begin{proof}\leanok 233 | \uses{} 234 | Uses no previous result 235 | \end{proof} 236 | 237 | \begin{lemma} \label{lem29} \lean{lemma29} 238 | For any $\eps>0$ and $N,r\ge1$, we have 239 | \[ 240 | \sum_{n\in \mathcal{R}}\frac{1}{n^{\eps}} \ge \frac{|\mathcal{R}|}{N^\eps}, 241 | \] 242 | \end{lemma} 243 | \begin{proof}\leanok 244 | \uses{def25, lem27, lem28} 245 | Uses lemma \ref{lem27} with $\mathcal S=\mathcal R$, and lemma \ref{lem28}. 246 | \end{proof} 247 | 248 | \begin{lemma} \label{lem30} 249 | Let $\eps>0$ and $N\ge1$. If $r=p_1p_2$, then $\mathcal{R} = \mathcal{R}(r,N)$ satisfies 250 | \[ 251 | \sum_{n\in \mathcal{R}}\frac{1}{n^{\eps}} 252 | \le \sum_{a_1\ge1}\sum_{a_2\ge1} \frac{1}{(p_1^{a_1}p_2^{a_2})^\eps} 253 | \] 254 | \end{lemma} 255 | \begin{proof}\leanok 256 | \uses{lem26} 257 | Uses lemma \ref{lem26}. 258 | \end{proof} 259 | 260 | \begin{lemma} \label{lem31} \lean{lemma31} 261 | Let $\eps>0$ and $N\ge1$. If $r=p_1\cdots p_k$, then $\mathcal{R} = \mathcal{R}(r,N)$ satisfies 262 | \[ 263 | \sum_{n\in \mathcal{R}}\frac{1}{n^{\eps}} 264 | \le \sum_{a_1,\ldots, a_k\ge1} \frac{1}{(p_1^{a_1}\cdots p_k^{a_k})^\eps} 265 | \] 266 | \end{lemma} 267 | \begin{proof}\leanok 268 | \uses{lem26} 269 | Uses lemma \ref{lem26}. 270 | \end{proof} 271 | 272 | \begin{lemma} \label{lem32} \lean{lemma32} 273 | Let $\eps>0$ and $N\ge1$. Then 274 | \[ 275 | \sum_{a_1\ge1}\sum_{a_2\ge1} \frac{1}{(p_1^{a_1}p_2^{a_2})^\eps} = \Bigl(\sum_{a_1\ge1} \frac{1}{p_1^{\eps a_1}}\Bigr) \Bigl(\sum_{a_2\ge1} \frac{1}{p_2^{\eps a_2}}\Bigr) 276 | \] 277 | \end{lemma} 278 | \begin{proof}\leanok 279 | \uses{} 280 | Uses no previous result 281 | \end{proof} 282 | 283 | \begin{lemma} \label{lem33} \lean{lemma33} 284 | Let $\eps>0$ and $p_1<\cdots0$ and let $p_i\ge2$ be a prime. Then 305 | \[ 306 | \sum_{a_i\ge1} \frac{1}{p_i^{\eps a_i}} = \frac{1}{p_i^\eps-1}. 307 | \] 308 | \end{lemma} 309 | \begin{proof}\leanok 310 | \uses{lem34} 311 | Uses lemma \ref{lem34} 312 | \end{proof} 313 | 314 | \begin{lemma} \label{lem36} \lean{lemma36} 315 | Let $\eps>0$ and let $p_1<\cdots0$ and let $p_1<\cdots0$. If $p_i^{\eps}\ge 2$ then $\frac{1}{p_i^{\eps}-1}\le 1$. 340 | \end{lemma} 341 | \begin{proof}\leanok 342 | \uses{} 343 | Uses no previous result 344 | \end{proof} 345 | 346 | \begin{lemma} \label{lem39} \lean{lemma39} 347 | Let $\eps>0$ and let $p_1<\cdots0$. Then $\frac{1}{2^{\eps}-1} \le \frac{2}{\eps}$. 359 | \end{lemma} 360 | \begin{proof}\leanok 361 | \uses{lem2} 362 | Uses lemma \ref{lem2} 363 | \end{proof} 364 | 365 | \begin{lemma} \label{lem41} \lean{lemma41} 366 | Let $p_i\ge2$, $a_i\ge1$ and $\eps>0$. If $p_i^{\eps}< 2$ then $\frac{1}{p_i^{\eps}-1} \le \frac{2}{\eps}$. 367 | \end{lemma} 368 | \begin{proof}\leanok 369 | \uses{lem40} 370 | Uses $p_i\ge2$ and lemma \ref{lem40} 371 | \end{proof} 372 | 373 | \begin{lemma} \label{lem42} \lean{lemma42} 374 | Let $\eps>0$ and let $p_1<\cdots0$ and let $p_1<\cdots0$ and let $p_1<\cdots0$ and $N\ge1$. Then $(2/\eps)^{2^{1/\eps}} \le O_\eps(1)$. 409 | \end{lemma} 410 | \begin{proof}\leanok 411 | \uses{lem14} 412 | Uses lemma \ref{lem14}. 413 | \end{proof} 414 | 415 | 416 | \begin{lemma} \label{lem46} \lean{lemma46} 417 | For any $\eps>0$ and $N,r\ge1$, we have 418 | \[ 419 | \frac{|\mathcal{R}|}{N^\eps} \le O_\eps(1). 420 | \] 421 | \end{lemma} 422 | \begin{proof}\leanok 423 | \uses{lem29, lem31, lem33, lem36, lem44, lem45} 424 | Uses lemmas \ref{lem29}, \ref{lem31}, \ref{lem33}, \ref{lem36}, \ref{lem44}, \ref{lem45}. 425 | \end{proof} 426 | 427 | \begin{lemma} \label{lem47} \lean{lemma47} 428 | Let $\eps>0$ and $1\le r\le N$. Then we have 429 | \[ 430 | |\mathcal{R}(r,N)| \le O_\eps(N^\eps) 431 | \] 432 | \end{lemma} 433 | \begin{proof}\leanok 434 | \uses{lem46} 435 | Uses lemma \ref{lem46} 436 | \end{proof} 437 | 438 | 439 | \begin{theorem} \label{lem:radical} \label{thm48} \lean{theorem48} 440 | Let $1\le r\le N$. Then we have 441 | \[ 442 | |\mathcal{R}(r,N)| \le N^{o(1)}. 443 | \] 444 | \end{theorem} 445 | \begin{proof}\leanok 446 | \uses{lem47} 447 | Uses lemma \ref{lem47} and definition of $o(1)$ 448 | \end{proof} 449 | 450 | \begin{lemma} \label{lem49} \lean{lemma49} 451 | Let $N\ge1$ and $0<\lambda<1$. We have 452 | \[ 453 | \big|\big\{n\le N : \rad(n) \le N^{\lambda}\big\}\big| = \sum_{1\le r\le N^{\lambda}}\sum_{\substack{1\le n\le N \\ \rad(n) = r}}1 454 | \] 455 | \end{lemma} 456 | \begin{proof}\leanok 457 | \uses{lem27} 458 | Uses no previous results 459 | \end{proof} 460 | 461 | \begin{lemma} \label{lem50} \lean{lemma50} 462 | Let $N\ge1$ and $0<\lambda<1$. We have 463 | \[ 464 | \big|\big\{n\le N : \rad(n) \le N^{\lambda}\big\}\big| = \sum_{1\le r\le N^{\lambda}}|\mathcal R(r,N)| 465 | \] 466 | \end{lemma} 467 | \begin{proof}\leanok 468 | \uses{lem49, def25, lem27} 469 | Uses lemma \ref{lem49} and definition \ref{def25}. 470 | \end{proof} 471 | 472 | \begin{lemma} \label{lem51} \lean{lemma51} 473 | Let $N\ge1$ and $0<\lambda<1$. We have 474 | \[ 475 | \big|\big\{n\le N : \rad(n) \le N^{\lambda}\big\}\big| \le \sum_{1\le r\le N^{\lambda}}N^{o(1)} 476 | \] 477 | \end{lemma} 478 | \begin{proof}\leanok 479 | \uses{lem50, thm48} 480 | Uses lemma \ref{lem50} and theorem \ref{thm48}. 481 | \end{proof} 482 | 483 | 484 | \begin{theorem}\label{cor:rad} \label{thm52} \lean{corollary52} 485 | Let $N\ge1$ and $0<\lambda<1$. We have 486 | \[ 487 | \big|\big\{n\le N : \rad(n) \le N^{\lambda}\big\}\big| \ \le \ N^{\lambda+o(1)}. 488 | \] 489 | \end{theorem} 490 | \begin{proof}\leanok 491 | \uses{lem51} 492 | Uses lemma \ref{lem51} 493 | \end{proof} 494 | 495 | \begin{definition}[Exceptional set] \label{def53} \lean{exceptionalSet}\leanok 496 | Let $N\ge1$ and $\eps>0$. Define the exceptional set 497 | \[ 498 | \mathcal E(N) \; = \; \bigg\{(a,b,c)\in \{1,\ldots,N\}^3 \;:\; \gcd(a,b)=1, a+b=c, \ \rad(abc) \, < \, c^{1-\eps} 499 | \bigg\}. 500 | \] 501 | \end{definition} 502 | 503 | \begin{lemma} \label{lem54} \lean{lemma54} 504 | Let $a,b\ge1$ be integers with $\gcd(a,bc)=1$ and $\gcd(b,c)=1$. If $\rad(ab)$, $\rad(ac)$, $\rad(bc)$ all exceed $c^{\frac{2}{3}(1-\eps)}$, then 505 | \[ 506 | c^{2-2\eps}\le\rad(ab) \rad(ac)\rad(bc) = (\rad(abc))^2. 507 | \] 508 | \end{lemma} 509 | \begin{proof}\leanok 510 | \uses{lem23} 511 | Uses lemma \ref{lem23} 512 | \end{proof} 513 | 514 | \begin{lemma} \label{lem55} \lean{lemma55} 515 | Let $a,b\ge1$ be integers with $\gcd(a,bc)=1$ and $\gcd(b,c)=1$. If $\rad(ab)$, $\rad(ac)$, $\rad(bc)$ all exceed $c^{\frac{2}{3}(1-\eps)}$, then $c^{1-\eps}\le\rad(abc)$. 516 | \end{lemma} 517 | \begin{proof}\leanok 518 | \uses{lem54} 519 | Uses lemma \ref{lem54} 520 | \end{proof} 521 | 522 | 523 | \begin{lemma} \label{lem56} \lean{lemma56} 524 | Let $(a,b,c)\in \mathcal E(N)$. Then either $\rad(ab)0$ and $N\ge1$. We have 552 | \[ 553 | \underset{\substack{1\le a,b,c\le N\\ \gcd(a,b)=1\\ \rad(abc) \ < \ c^{1-\eps} \\ a+b=c}}{\sum\sum\sum} 1 554 | \ \le \ \sum_{r\le N^{2/3-\eps}}\underset{\substack{1\le a,b,c\le N\\ \gcd(a,b)=1\\ r=\rad(abc) \\ a+b=c}}{\sum\sum\sum} 1 555 | \] 556 | \end{lemma} 557 | \begin{proof}\leanok 558 | \uses{def53, lem27, lem57} 559 | Uses no previous results 560 | \end{proof} 561 | 562 | \begin{lemma} \label{lem60} \lean{lemma60} 563 | Let $\eps>0$ and $r,N\ge1$. We have 564 | \[ 565 | \underset{\substack{1\le a,b,c\le N\\ \gcd(a,b)=1\\ r=\rad(abc) \\ a+b=c}}{\sum\sum\sum} 1 566 | \ \le \ 3\underset{\substack{1\le x,y\le N\\ \gcd(x,y)=1\\r=\rad(xy)}}{\sum\sum} 1 567 | \] 568 | \end{lemma} 569 | \begin{proof}\leanok 570 | \uses{lem57} 571 | Uses lemma \ref{lem57} 572 | \end{proof} 573 | 574 | \begin{lemma} \label{lem61} \lean{lemma61} 575 | Let $\eps>0$ and $r,N\ge1$. We have 576 | \[ 577 | \underset{\substack{1\le x,y\le N\\ \gcd(x,y)=1\\r=\rad(xy)}}{\sum\sum} 1 578 | \ \le \ \sum_{\substack{1\le n\le N^2\\r=\rad(n)}} \tau(n). 579 | \] 580 | \end{lemma} 581 | \begin{proof}\leanok 582 | \uses{def4} 583 | Uses definition \ref{def4} with $n=xy$ 584 | \end{proof} 585 | 586 | \begin{lemma} \label{lem62} \lean{lemma62} 587 | We have 588 | \[ 589 | \sum_{\substack{1\le n\le N^2\\r=\rad(n)}} \tau(n) 590 | \le \sum_{\substack{1\le n\le N^2\\r=\rad(n)}} N^{o(1)} 591 | \] 592 | \end{lemma} 593 | \begin{proof}\leanok 594 | \uses{thm18} 595 | Uses theorem \ref{thm18}, and that $n\le N$ 596 | \end{proof} 597 | 598 | \begin{lemma} \label{lem63} \lean{lemma63} 599 | We have 600 | \[ 601 | 3\sum_{r\le N^{2/3-\eps}}\underset{\substack{1\le x,y\le N\\ \gcd(x,y)=1\\r=\rad(xy)}}{\sum\sum} 1 602 | \ \le \ \sum_{r\le N^{2/3-\eps}}\sum_{\substack{1\le n\le N^2\\r=\rad(n)}} N^{o(1)} 603 | \] 604 | \end{lemma} 605 | \begin{proof}\leanok 606 | \uses{lem61, lem62} 607 | Uses lemmas \ref{lem61}, \ref{lem62} 608 | \end{proof} 609 | 610 | \begin{lemma} \label{lem64} \lean{lemma64} 611 | We have 612 | \[ 613 | \sum_{r\le N^{2/3-\eps}}\sum_{\substack{1\le n\le N^2\\r=\rad(n)}}1 = \big|\big\{n\le N^2 : \rad(n) \le N^{2/3-\eps}\big\}\big|. 614 | \] 615 | \end{lemma} 616 | \begin{proof}\leanok 617 | \uses{} 618 | Uses no previous results 619 | \end{proof} 620 | 621 | 622 | \begin{lemma} \label{lem65} \lean{lemma65} 623 | Let $N\ge1$ and $\eps>0$. We have 624 | \[ 625 | |\mathcal E(N)| \ \le \ N^{o(1)}\cdot\big|\big\{n\le N^2 : \rad(n) \le N^{2/3-\eps}\big\}\big| 626 | \] 627 | \end{lemma} 628 | \begin{proof}\leanok 629 | \uses{lem58, lem59, lem60, lem63, lem64} 630 | Uses lemmas \ref{lem58}, \ref{lem59}, \ref{lem60}, \ref{lem63}, \ref{lem64} 631 | \end{proof} 632 | 633 | \begin{lemma} \label{lem66} \lean{lemma66} 634 | Let $N\ge1$ and $\eps>0$. We have 635 | \[ 636 | \big|\big\{n\le N^2 : \rad(n) \le N^{2/3-\eps}\big\}\big| \ \le \ N^{2/3-\eps+o(1)} 637 | \] 638 | \end{lemma} 639 | \begin{proof}\leanok 640 | \uses{def25, thm52} 641 | Uses definition \ref{def25}, theorem \ref{thm52} with $N^2$ and $\lambda=2/3-\eps$ 642 | \end{proof} 643 | 644 | \begin{theorem}\label{thm:deBr} \label{thm67} \lean{theorem67} 645 | We have $|\mathcal E(N)| \le O(N^{2/3})$. 646 | \end{theorem} 647 | \begin{proof}\leanok 648 | \uses{lem65, lem66} 649 | Uses lemmas \ref{lem65}, \ref{lem66} 650 | \end{proof} 651 | --------------------------------------------------------------------------------