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<H1><A NAME="SECTION0017300000000000000000">
15.3 <tt class="module">mpz</tt> --
GNU arbitrary magnitude integers</A>
</H1>
<P>
<P>
This is an optional module. It is only available when Python is
configured to include it, which requires that the GNU MP software is
installed.
<P>
This module implements the interface to part of the GNU MP library,
which defines arbitrary precision integer and rational number
arithmetic routines. Only the interfaces to the <i>integer</i>
(<tt class="function">mpz_*()</tt>) routines are provided. If not stated
otherwise, the description in the GNU MP documentation can be applied.
<P>
Support for rational numbers can be
implemented in Python. For an example, see the
<tt class="module">Rat</tt> module, provided as
<span class="file">Demos/classes/Rat.py</span> in the Python source distribution.
<P>
In general, <i class="dfn">mpz</i>-numbers can be used just like other standard
Python numbers, e.g., you can use the built-in operators like <code>+</code>,
<code>*</code>, etc., as well as the standard built-in functions like
<tt class="function">abs()</tt>, <tt class="function">int()</tt>, ..., <tt class="function">divmod()</tt>,
<tt class="function">pow()</tt>. <b>Please note:</b> the <i>bitwise-xor</i>
operation has been implemented as a bunch of <i>and</i>s,
<i>invert</i>s and <i>or</i>s, because the library lacks an
<tt class="cfunction">mpz_xor()</tt> function, and I didn't need one.
<P>
You create an mpz-number by calling the function <tt class="function">mpz()</tt> (see
below for an exact description). An mpz-number is printed like this:
<code>mpz(<var>value</var>)</code>.
<P>
<dl><dt><b><a name='l2h-2965'><tt class='function'>mpz</tt></a></b> (<var>value</var>)
<dd>
Create a new mpz-number. <var>value</var> can be an integer, a long,
another mpz-number, or even a string. If it is a string, it is
interpreted as an array of radix-256 digits, least significant digit
first, resulting in a positive number. See also the <tt class="method">binary()</tt>
method, described below.
</dl>
<P>
<dl><dt><b><a name='l2h-2966'><tt>MPZType</tt></a></b>
<dd>
The type of the objects returned by <tt class="function">mpz()</tt> and most other
functions in this module.
</dl>
<P>
A number of <i>extra</i> functions are defined in this module. Non
mpz-arguments are converted to mpz-values first, and the functions
return mpz-numbers.
<P>
<dl><dt><b><a name='l2h-2967'><tt class='function'>powm</tt></a></b> (<var>base, exponent, modulus</var>)
<dd>
Return <code>pow(<var>base</var>, <var>exponent</var>) % <var>modulus</var></code>. If
<code><var>exponent</var> == 0</code>, return <code>mpz(1)</code>. In contrast to the
C library function, this version can handle negative exponents.
</dl>
<P>
<dl><dt><b><a name='l2h-2968'><tt class='function'>gcd</tt></a></b> (<var>op1, op2</var>)
<dd>
Return the greatest common divisor of <var>op1</var> and <var>op2</var>.
</dl>
<P>
<dl><dt><b><a name='l2h-2969'><tt class='function'>gcdext</tt></a></b> (<var>a, b</var>)
<dd>
Return a tuple <code>(<var>g</var>, <var>s</var>, <var>t</var>)</code>, such that
<code><var>a</var>*<var>s</var> + <var>b</var>*<var>t</var> == <var>g</var> == gcd(<var>a</var>, <var>b</var>)</code>.
</dl>
<P>
<dl><dt><b><a name='l2h-2970'><tt class='function'>sqrt</tt></a></b> (<var>op</var>)
<dd>
Return the square root of <var>op</var>. The result is rounded towards zero.
</dl>
<P>
<dl><dt><b><a name='l2h-2971'><tt class='function'>sqrtrem</tt></a></b> (<var>op</var>)
<dd>
Return a tuple <code>(<var>root</var>, <var>remainder</var>)</code>, such that
<code><var>root</var>*<var>root</var> + <var>remainder</var> == <var>op</var></code>.
</dl>
<P>
<dl><dt><b><a name='l2h-2972'><tt class='function'>divm</tt></a></b> (<var>numerator, denominator, modulus</var>)
<dd>
Returns a number <var>q</var> such that
<code><var>q</var> * <var>denominator</var> % <var>modulus</var> ==
<var>numerator</var></code>. One could also implement this function in Python,
using <tt class="function">gcdext()</tt>.
</dl>
<P>
An mpz-number has one method:
<P>
<dl><dt><b><a name='l2h-2973'><tt class='method'>binary</tt></a></b> ()
<dd>
Convert this mpz-number to a binary string, where the number has been
stored as an array of radix-256 digits, least significant digit first.
<P>
The mpz-number must have a value greater than or equal to zero,
otherwise <tt class="exception">ValueError</tt> will be raised.
</dl>
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