Team Ai
Apppublic

parthtamu/rag-code-assistant

sourceHugging Faceupdated 7mo agoView on Hugging Face
0likes
math.integer.html418 linesDownload Raw Back to docs
1<!DOCTYPE html>2 3<html lang="en" data-content_root="../">4  <head>5    <meta charset="utf-8" />6    <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />7<meta property="og:title" content="math.integer — integer-specific mathematics functions" />8<meta property="og:type" content="website" />9<meta property="og:url" content="https://docs.python.org/3/library/math.integer.html" />10<meta property="og:site_name" content="Python documentation" />11<meta property="og:description" content="This module provides access to the mathematical functions defined for integer arguments. These functions accept integers and objects that implement the__index__() method which is used to convert th..." />12<meta property="og:image:width" content="1146" />13<meta property="og:image:height" content="600" />14<meta property="og:image" content="https://docs.python.org/3.15/_images/social_previews/summary_library_math.integer_4cf6f44c.png" />15<meta property="og:image:alt" content="This module provides access to the mathematical functions defined for integer arguments. These functions accept integers and objects that implement the__index__() method which is used to convert th..." />16<meta name="description" content="This module provides access to the mathematical functions defined for integer arguments. These functions accept integers and objects that implement the__index__() method which is used to convert th..." />17<meta name="twitter:card" content="summary_large_image" />18<meta name="theme-color" content="#3776ab">19 20    <title>math.integer — integer-specific mathematics functions &#8212; Python 3.15.0a6 documentation</title><meta name="viewport" content="width=device-width, initial-scale=1.0">21    22    <link rel="stylesheet" type="text/css" href="../_static/pygments.css?v=b86133f3" />23    <link rel="stylesheet" type="text/css" href="../_static/classic.css?v=234b1a7c" />24    <link rel="stylesheet" type="text/css" href="../_static/pydoctheme.css?v=89a2f22a" />25    <link rel="stylesheet" type="text/css" href="../_static/profiling-sampling-visualization.css?v=0c2600ae" />26    <link id="pygments_dark_css" media="(prefers-color-scheme: dark)" rel="stylesheet" type="text/css" href="../_static/pygments_dark.css?v=5349f25f" />27    28    <script src="../_static/documentation_options.js?v=6b7c9ff5"></script>29    <script src="../_static/doctools.js?v=9bcbadda"></script>30    <script src="../_static/sphinx_highlight.js?v=dc90522c"></script>31    <script src="../_static/profiling-sampling-visualization.js?v=9811ed04"></script>32    33    <script src="../_static/sidebar.js"></script>34    35    <link rel="search" type="application/opensearchdescription+xml"36          title="Search within Python 3.15.0a6 documentation"37          href="../_static/opensearch.xml"/>38    <link rel="author" title="About these documents" href="../about.html" />39    <link rel="index" title="Index" href="../genindex.html" />40    <link rel="search" title="Search" href="../search.html" />41    <link rel="copyright" title="Copyright" href="../copyright.html" />42    <link rel="next" title="cmath — Mathematical functions for complex numbers" href="cmath.html" />43    <link rel="prev" title="math — Mathematical functions" href="math.html" />44    45      46      <script defer file-types="bz2,epub,zip" data-domain="docs.python.org" src="https://analytics.python.org/js/script.file-downloads.outbound-links.js"></script>47      48      <link rel="canonical" href="https://docs.python.org/3/library/math.integer.html">49      50    51 52    53    <style>54      @media only screen {55        table.full-width-table {56            width: 100%;57        }58      }59    </style>60<link rel="stylesheet" href="../_static/pydoctheme_dark.css" media="(prefers-color-scheme: dark)" id="pydoctheme_dark_css">61    <link rel="shortcut icon" type="image/png" href="../_static/py.svg">62            <script type="text/javascript" src="../_static/copybutton.js"></script>63            <script type="text/javascript" src="../_static/menu.js"></script>64            <script type="text/javascript" src="../_static/search-focus.js"></script>65            <script type="text/javascript" src="../_static/themetoggle.js"></script> 66            <script type="text/javascript" src="../_static/rtd_switcher.js"></script>67            <meta name="readthedocs-addons-api-version" content="1">68 69  </head>70<body>71<div class="mobile-nav">72    <input type="checkbox" id="menuToggler" class="toggler__input" aria-controls="navigation"73           aria-pressed="false" aria-expanded="false" role="button" aria-label="Menu">74    <nav class="nav-content" role="navigation">75        <label for="menuToggler" class="toggler__label">76            <span></span>77        </label>78        <span class="nav-items-wrapper">79            <a href="https://www.python.org/" class="nav-logo">80                <img src="../_static/py.svg" alt="Python logo">81            </a>82            <span class="version_switcher_placeholder"></span>83            <form role="search" class="search" action="../search.html" method="get">84                <svg xmlns="http://www.w3.org/2000/svg" width="20" height="20" viewBox="0 0 24 24" class="search-icon">85                    <path fill-rule="nonzero" fill="currentColor" d="M15.5 14h-.79l-.28-.27a6.5 6.5 0 001.48-5.34c-.47-2.78-2.79-5-5.59-5.34a6.505 6.505 0 00-7.27 7.27c.34 2.8 2.56 5.12 5.34 5.59a6.5 6.5 0 005.34-1.48l.27.28v.79l4.25 4.25c.41.41 1.08.41 1.49 0 .41-.41.41-1.08 0-1.49L15.5 14zm-6 0C7.01 14 5 11.99 5 9.5S7.01 5 9.5 5 14 7.01 14 9.5 11.99 14 9.5 14z"></path>86                </svg>87                <input placeholder="Quick search" aria-label="Quick search" type="search" name="q">88                <input type="submit" value="Go">89            </form>90        </span>91    </nav>92    <div class="menu-wrapper">93        <nav class="menu" role="navigation" aria-label="main navigation">94            <div class="language_switcher_placeholder"></div>95            96<label class="theme-selector-label">97    Theme98    <select class="theme-selector" oninput="activateTheme(this.value)">99        <option value="auto" selected>Auto</option>100        <option value="light">Light</option>101        <option value="dark">Dark</option>102    </select>103</label>104  <div>105    <h4>Previous topic</h4>106    <p class="topless"><a href="math.html"107                          title="previous chapter"><code class="xref py py-mod docutils literal notranslate"><span class="pre">math</span></code> — Mathematical functions</a></p>108  </div>109  <div>110    <h4>Next topic</h4>111    <p class="topless"><a href="cmath.html"112                          title="next chapter"><code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code> — Mathematical functions for complex numbers</a></p>113  </div>114  <script>115    document.addEventListener('DOMContentLoaded', () => {116        const title = document.querySelector('meta[property="og:title"]').content;117        const elements = document.querySelectorAll('.improvepage');118        const pageurl = window.location.href.split('?')[0];119        elements.forEach(element => {120            const url = new URL(element.href.split('?')[0].replace("-nojs", ""));121            url.searchParams.set('pagetitle', title);122            url.searchParams.set('pageurl', pageurl);123            url.searchParams.set('pagesource', "library/math.integer.rst");124            element.href = url.toString();125        });126    });127  </script>128  <div role="note" aria-label="source link">129    <h3>This page</h3>130    <ul class="this-page-menu">131      <li><a href="../bugs.html">Report a bug</a></li>132      <li><a class="improvepage" href="../improve-page-nojs.html">Improve this page</a></li>133      <li>134        <a href="https://github.com/python/cpython/blob/main/Doc/library/math.integer.rst?plain=1"135            rel="nofollow">Show source136        </a>137      </li>138      139    </ul>140  </div>141        </nav>142    </div>143</div>144 145  146    <div class="related" role="navigation" aria-label="Related">147      <h3>Navigation</h3>148      <ul>149        <li class="right" style="margin-right: 10px">150          <a href="../genindex.html" title="General Index"151             accesskey="I">index</a></li>152        <li class="right" >153          <a href="../py-modindex.html" title="Python Module Index"154             >modules</a> |</li>155        <li class="right" >156          <a href="cmath.html" title="cmath — Mathematical functions for complex numbers"157             accesskey="N">next</a> |</li>158        <li class="right" >159          <a href="math.html" title="math — Mathematical functions"160             accesskey="P">previous</a> |</li>161 162          <li><img src="../_static/py.svg" alt="Python logo" style="vertical-align: middle; margin-top: -1px"></li>163          <li><a href="https://www.python.org/">Python</a> &#187;</li>164          <li class="switchers">165            <div class="language_switcher_placeholder"></div>166            <div class="version_switcher_placeholder"></div>167          </li>168          <li>169              170          </li>171    <li id="cpython-language-and-version">172      <a href="../index.html">3.15.0a6 Documentation</a> &#187;173    </li>174 175          <li class="nav-item nav-item-1"><a href="index.html" >The Python Standard Library</a> &#187;</li>176          <li class="nav-item nav-item-2"><a href="numeric.html" accesskey="U">Numeric and Mathematical Modules</a> &#187;</li>177        <li class="nav-item nav-item-this"><a href=""><code class="xref py py-mod docutils literal notranslate"><span class="pre">math.integer</span></code> — integer-specific mathematics functions</a></li>178                <li class="right">179                    180 181    <div class="inline-search" role="search">182        <form class="inline-search" action="../search.html" method="get">183          <input placeholder="Quick search" aria-label="Quick search" type="search" name="q" id="search-box">184          <input type="submit" value="Go">185        </form>186    </div>187                     |188                </li>189            <li class="right">190<label class="theme-selector-label">191    Theme192    <select class="theme-selector" oninput="activateTheme(this.value)">193        <option value="auto" selected>Auto</option>194        <option value="light">Light</option>195        <option value="dark">Dark</option>196    </select>197</label> |</li>198            199      </ul>200    </div>    201 202    <div class="document">203      <div class="documentwrapper">204        <div class="bodywrapper">205          <div class="body" role="main">206            207  <section id="module-math.integer">208<span id="math-integer-integer-specific-mathematics-functions"></span><h1><code class="xref py py-mod docutils literal notranslate"><span class="pre">math.integer</span></code> — integer-specific mathematics functions<a class="headerlink" href="#module-math.integer" title="Link to this heading">¶</a></h1>209<div class="versionadded">210<p><span class="versionmodified added">Added in version 3.15.</span></p>211</div>212<hr class="docutils" />213<p>This module provides access to the mathematical functions defined for integer arguments.214These functions accept integers and objects that implement the215<a class="reference internal" href="../reference/datamodel.html#object.__index__" title="object.__index__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__index__()</span></code></a> method which is used to convert the object to an integer216number.</p>217<p>The following functions are provided by this module.  All return values are218computed exactly and are integers.</p>219<dl class="py function">220<dt class="sig sig-object py" id="math.integer.comb">221<span class="sig-prename descclassname"><span class="pre">math.integer.</span></span><span class="sig-name descname"><span class="pre">comb</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</span></span></em>, <em class="sig-param"><span class="positional-only-separator o"><abbr title="Positional-only parameter separator (PEP 570)"><span class="pre">/</span></abbr></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.integer.comb" title="Link to this definition">¶</a></dt>222<dd><p>Return the number of ways to choose <em>k</em> items from <em>n</em> items without repetition223and without order.</p>224<p>Evaluates to <code class="docutils literal notranslate"><span class="pre">n!</span> <span class="pre">/</span> <span class="pre">(k!</span> <span class="pre">*</span> <span class="pre">(n</span> <span class="pre">-</span> <span class="pre">k)!)</span></code> when <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">&lt;=</span> <span class="pre">n</span></code> and evaluates225to zero when <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">&gt;</span> <span class="pre">n</span></code>.</p>226<p>Also called the binomial coefficient because it is equivalent227to the coefficient of k-th term in polynomial expansion of228<code class="docutils literal notranslate"><span class="pre">(1</span> <span class="pre">+</span> <span class="pre">x)ⁿ</span></code>.</p>229<p>Raises <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a> if either of the arguments are negative.</p>230</dd></dl>231 232<dl class="py function">233<dt class="sig sig-object py" id="math.integer.factorial">234<span class="sig-prename descclassname"><span class="pre">math.integer.</span></span><span class="sig-name descname"><span class="pre">factorial</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em>, <em class="sig-param"><span class="positional-only-separator o"><abbr title="Positional-only parameter separator (PEP 570)"><span class="pre">/</span></abbr></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.integer.factorial" title="Link to this definition">¶</a></dt>235<dd><p>Return factorial of the nonnegative integer <em>n</em>.</p>236</dd></dl>237 238<dl class="py function">239<dt class="sig sig-object py" id="math.integer.gcd">240<span class="sig-prename descclassname"><span class="pre">math.integer.</span></span><span class="sig-name descname"><span class="pre">gcd</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">integers</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.integer.gcd" title="Link to this definition">¶</a></dt>241<dd><p>Return the greatest common divisor of the specified integer arguments.242If any of the arguments is nonzero, then the returned value is the largest243positive integer that is a divisor of all arguments.  If all arguments244are zero, then the returned value is <code class="docutils literal notranslate"><span class="pre">0</span></code>.  <code class="docutils literal notranslate"><span class="pre">gcd()</span></code> without arguments245returns <code class="docutils literal notranslate"><span class="pre">0</span></code>.</p>246</dd></dl>247 248<dl class="py function">249<dt class="sig sig-object py" id="math.integer.isqrt">250<span class="sig-prename descclassname"><span class="pre">math.integer.</span></span><span class="sig-name descname"><span class="pre">isqrt</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em>, <em class="sig-param"><span class="positional-only-separator o"><abbr title="Positional-only parameter separator (PEP 570)"><span class="pre">/</span></abbr></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.integer.isqrt" title="Link to this definition">¶</a></dt>251<dd><p>Return the integer square root of the nonnegative integer <em>n</em>. This is the252floor of the exact square root of <em>n</em>, or equivalently the greatest integer253<em>a</em> such that <em>a</em>² ≤ <em>n</em>.</p>254<p>For some applications, it may be more convenient to have the least integer255<em>a</em> such that <em>n</em> ≤ <em>a</em>², or in other words the ceiling of256the exact square root of <em>n</em>. For positive <em>n</em>, this can be computed using257<code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">=</span> <span class="pre">1</span> <span class="pre">+</span> <span class="pre">isqrt(n</span> <span class="pre">-</span> <span class="pre">1)</span></code>.</p>258</dd></dl>259 260<dl class="py function">261<dt class="sig sig-object py" id="math.integer.lcm">262<span class="sig-prename descclassname"><span class="pre">math.integer.</span></span><span class="sig-name descname"><span class="pre">lcm</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">integers</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.integer.lcm" title="Link to this definition">¶</a></dt>263<dd><p>Return the least common multiple of the specified integer arguments.264If all arguments are nonzero, then the returned value is the smallest265positive integer that is a multiple of all arguments.  If any of the arguments266is zero, then the returned value is <code class="docutils literal notranslate"><span class="pre">0</span></code>.  <code class="docutils literal notranslate"><span class="pre">lcm()</span></code> without arguments267returns <code class="docutils literal notranslate"><span class="pre">1</span></code>.</p>268</dd></dl>269 270<dl class="py function">271<dt class="sig sig-object py" id="math.integer.perm">272<span class="sig-prename descclassname"><span class="pre">math.integer.</span></span><span class="sig-name descname"><span class="pre">perm</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="positional-only-separator o"><abbr title="Positional-only parameter separator (PEP 570)"><span class="pre">/</span></abbr></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.integer.perm" title="Link to this definition">¶</a></dt>273<dd><p>Return the number of ways to choose <em>k</em> items from <em>n</em> items274without repetition and with order.</p>275<p>Evaluates to <code class="docutils literal notranslate"><span class="pre">n!</span> <span class="pre">/</span> <span class="pre">(n</span> <span class="pre">-</span> <span class="pre">k)!</span></code> when <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">&lt;=</span> <span class="pre">n</span></code> and evaluates276to zero when <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">&gt;</span> <span class="pre">n</span></code>.</p>277<p>If <em>k</em> is not specified or is <code class="docutils literal notranslate"><span class="pre">None</span></code>, then <em>k</em> defaults to <em>n</em>278and the function returns <code class="docutils literal notranslate"><span class="pre">n!</span></code>.</p>279<p>Raises <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a> if either of the arguments are negative.</p>280</dd></dl>281 282</section>283 284 285            <div class="clearer"></div>286          </div>287        </div>288      </div>289      <div class="sphinxsidebar" role="navigation" aria-label="Main">290        <div class="sphinxsidebarwrapper">291  <div>292    <h4>Previous topic</h4>293    <p class="topless"><a href="math.html"294                          title="previous chapter"><code class="xref py py-mod docutils literal notranslate"><span class="pre">math</span></code> — Mathematical functions</a></p>295  </div>296  <div>297    <h4>Next topic</h4>298    <p class="topless"><a href="cmath.html"299                          title="next chapter"><code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code> — Mathematical functions for complex numbers</a></p>300  </div>301  <script>302    document.addEventListener('DOMContentLoaded', () => {303        const title = document.querySelector('meta[property="og:title"]').content;304        const elements = document.querySelectorAll('.improvepage');305        const pageurl = window.location.href.split('?')[0];306        elements.forEach(element => {307            const url = new URL(element.href.split('?')[0].replace("-nojs", ""));308            url.searchParams.set('pagetitle', title);309            url.searchParams.set('pageurl', pageurl);310            url.searchParams.set('pagesource', "library/math.integer.rst");311            element.href = url.toString();312        });313    });314  </script>315  <div role="note" aria-label="source link">316    <h3>This page</h3>317    <ul class="this-page-menu">318      <li><a href="../bugs.html">Report a bug</a></li>319      <li><a class="improvepage" href="../improve-page-nojs.html">Improve this page</a></li>320      <li>321        <a href="https://github.com/python/cpython/blob/main/Doc/library/math.integer.rst?plain=1"322            rel="nofollow">Show source323        </a>324      </li>325      326    </ul>327  </div>328        </div>329<div id="sidebarbutton" title="Collapse sidebar">330<span>«</span>331</div>332 333      </div>334      <div class="clearer"></div>335    </div>  336    <div class="related" role="navigation" aria-label="Related">337      <h3>Navigation</h3>338      <ul>339        <li class="right" style="margin-right: 10px">340          <a href="../genindex.html" title="General Index"341             >index</a></li>342        <li class="right" >343          <a href="../py-modindex.html" title="Python Module Index"344             >modules</a> |</li>345        <li class="right" >346          <a href="cmath.html" title="cmath — Mathematical functions for complex numbers"347             >next</a> |</li>348        <li class="right" >349          <a href="math.html" title="math — Mathematical functions"350             >previous</a> |</li>351 352          <li><img src="../_static/py.svg" alt="Python logo" style="vertical-align: middle; margin-top: -1px"></li>353          <li><a href="https://www.python.org/">Python</a> &#187;</li>354          <li class="switchers">355            <div class="language_switcher_placeholder"></div>356            <div class="version_switcher_placeholder"></div>357          </li>358          <li>359              360          </li>361    <li id="cpython-language-and-version">362      <a href="../index.html">3.15.0a6 Documentation</a> &#187;363    </li>364 365          <li class="nav-item nav-item-1"><a href="index.html" >The Python Standard Library</a> &#187;</li>366          <li class="nav-item nav-item-2"><a href="numeric.html" >Numeric and Mathematical Modules</a> &#187;</li>367        <li class="nav-item nav-item-this"><a href=""><code class="xref py py-mod docutils literal notranslate"><span class="pre">math.integer</span></code> — integer-specific mathematics functions</a></li>368                <li class="right">369                    370 371    <div class="inline-search" role="search">372        <form class="inline-search" action="../search.html" method="get">373          <input placeholder="Quick search" aria-label="Quick search" type="search" name="q" id="search-box">374          <input type="submit" value="Go">375        </form>376    </div>377                     |378                </li>379            <li class="right">380<label class="theme-selector-label">381    Theme382    <select class="theme-selector" oninput="activateTheme(this.value)">383        <option value="auto" selected>Auto</option>384        <option value="light">Light</option>385        <option value="dark">Dark</option>386    </select>387</label> |</li>388            389      </ul>390    </div>  391    <div class="footer">392    &copy; <a href="../copyright.html">Copyright</a> 2001 Python Software Foundation.393    <br>394    This page is licensed under the Python Software Foundation License Version 2.395    <br>396    Examples, recipes, and other code in the documentation are additionally licensed under the Zero Clause BSD License.397    <br>398    399      See <a href="/license.html">History and License</a> for more information.<br>400    401    402    <br>403 404    The Python Software Foundation is a non-profit corporation.405<a href="https://www.python.org/psf/donations/">Please donate.</a>406<br>407    <br>408      Last updated on Mar 10, 2026 (08:58 UTC).409    410      <a href="/bugs.html">Found a bug</a>?411    412    <br>413 414    Created using <a href="https://www.sphinx-doc.org/">Sphinx</a> 8.2.3.415    </div>416 417  </body>418</html>