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<section id="module-math.integer">
<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>
<div class="versionadded">
<p><span class="versionmodified added">Added in version 3.15.</span></p>
</div>
<hr class="docutils" />
<p>This module provides access to the mathematical functions defined for integer arguments.
These functions accept integers and objects that implement the
<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 integer
number.</p>
<p>The following functions are provided by this module. All return values are
computed exactly and are integers.</p>
<dl class="py function">
<dt class="sig sig-object py" id="math.integer.comb">
<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>
<dd><p>Return the number of ways to choose <em>k</em> items from <em>n</em> items without repetition
and without order.</p>
<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 evaluates
to zero when <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">></span> <span class="pre">n</span></code>.</p>
<p>Also called the binomial coefficient because it is equivalent
to the coefficient of k-th term in polynomial expansion of
<code class="docutils literal notranslate"><span class="pre">(1</span> <span class="pre">+</span> <span class="pre">x)ⁿ</span></code>.</p>
<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>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="math.integer.factorial">
<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>
<dd><p>Return factorial of the nonnegative integer <em>n</em>.</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="math.integer.gcd">
<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>
<dd><p>Return the greatest common divisor of the specified integer arguments.
If any of the arguments is nonzero, then the returned value is the largest
positive integer that is a divisor of all arguments. If all arguments
are 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 arguments
returns <code class="docutils literal notranslate"><span class="pre">0</span></code>.</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="math.integer.isqrt">
<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>
<dd><p>Return the integer square root of the nonnegative integer <em>n</em>. This is the
floor of the exact square root of <em>n</em>, or equivalently the greatest integer
<em>a</em> such that <em>a</em>² ≤ <em>n</em>.</p>
<p>For some applications, it may be more convenient to have the least integer
<em>a</em> such that <em>n</em> ≤ <em>a</em>², or in other words the ceiling of
the exact square root of <em>n</em>. For positive <em>n</em>, this can be computed using
<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>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="math.integer.lcm">
<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>
<dd><p>Return the least common multiple of the specified integer arguments.
If all arguments are nonzero, then the returned value is the smallest
positive integer that is a multiple of all arguments. If any of the arguments
is 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 arguments
returns <code class="docutils literal notranslate"><span class="pre">1</span></code>.</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="math.integer.perm">
<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>
<dd><p>Return the number of ways to choose <em>k</em> items from <em>n</em> items
without repetition and with order.</p>
<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 evaluates
to zero when <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">></span> <span class="pre">n</span></code>.</p>
<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>
and the function returns <code class="docutils literal notranslate"><span class="pre">n!</span></code>.</p>
<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>
</dd></dl>
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