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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"><=</span> <span class="pre">n</span></code> and evaluates225to zero when <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">></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"><=</span> <span class="pre">n</span></code> and evaluates276to zero when <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">></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; 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