# 15.3 `mpz` -- GNU arbitrary magnitude integers

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.

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 integer (`mpz_*()`) routines are provided. If not stated otherwise, the description in the GNU MP documentation can be applied.

Support for rational numbers can be implemented in Python. For an example, see the `Rat` module, provided as Demos/classes/Rat.py in the Python source distribution.

In general, mpz-numbers can be used just like other standard Python numbers, e.g., you can use the built-in operators like `+`, `*`, etc., as well as the standard built-in functions like `abs()`, `int()`, ..., `divmod()`, `pow()`. Please note: the bitwise-xor operation has been implemented as a bunch of ands, inverts and ors, because the library lacks an `mpz_xor()` function, and I didn't need one.

You create an mpz-number by calling the function `mpz()` (see below for an exact description). An mpz-number is printed like this: `mpz(value)`.

`mpz`(value)
  Create a new mpz-number. value 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 `binary()` method, described below.

`MPZType`
  The type of the objects returned by `mpz()` and most other functions in this module.

A number of extra functions are defined in this module. Non mpz-arguments are converted to mpz-values first, and the functions return mpz-numbers.

`powm`(base, exponent, modulus)
  Return `pow(base, exponent) % modulus`. If `exponent == 0`, return `mpz(1)`. In contrast to the C library function, this version can handle negative exponents.

`gcd`(op1, op2)
  Return the greatest common divisor of op1 and op2.

`gcdext`(a, b)
  Return a tuple `(g, s, t)`, such that `a*s + b*t == g == gcd(a, b)`.

`sqrt`(op)
  Return the square root of op. The result is rounded towards zero.

`sqrtrem`(op)
  Return a tuple `(root, remainder)`, such that `root*root + remainder == op`.

`divm`(numerator, denominator, modulus)
  Returns a number q such that `q * denominator % modulus == numerator`. One could also implement this function in Python, using `gcdext()`.

An mpz-number has one method:

`binary`()
  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.

  The mpz-number must have a value greater than or equal to zero, otherwise `ValueError` will be raised.