arctans.primes
==============

.. py:module:: arctans.primes

.. autoapi-nested-parse::

   Mathematical utility functions.



Attributes
----------

.. autoapisummary::

   arctans.primes.primes


Functions
---------

.. autoapisummary::

   arctans.primes.pfactors
   arctans.primes.largest_pfactor
   arctans.primes.is_prime
   arctans.primes.is_irreducible
   arctans.primes.is_gaussian_prime
   arctans.primes.is_gaussian_unit
   arctans.primes.complex_factorise


Module Contents
---------------

.. py:data:: primes
   :value: [2]


.. py:function:: pfactors(n: int) -> list[arctans.numbers.Integer]

   Get list of all prime factors of n.

   :param n: An integer

   :returns: A list of the prime factors of n, including factors multiple times when they appear more than once in the prime factorisation


.. py:function:: largest_pfactor(n: int | arctans.numbers.Integer) -> arctans.numbers.Integer

   Compute the largest prime factor of n.

   :param n: An integer

   :returns: The largest prime factor of n


.. py:function:: is_prime(n: arctans.numbers.Integer | int) -> bool

   Check if an integer is prime.

   :param n: An integer

   :returns: True if n is prime


.. py:function:: is_irreducible(n: arctans.numbers.Integer | int) -> bool

   Check if arctan(n) is irreducible.

   An arctan is irreducible iff it cannot be written as a
   weighted sum of integer arccotangents, or equivalently
   arctan(n) is irreducible iff the largest prime factor of
   1 + n**2 is greater than or equal to 2*n.

   :param n: An integer

   :returns: True if n is irreducible


.. py:function:: is_gaussian_prime(n: arctans.numbers.GaussianInteger) -> bool

   Check if n is a Gaussian prime.

   :param n: An integer

   :returns: True if n is a Gaussian prime


.. py:function:: is_gaussian_unit(n: arctans.numbers.GaussianInteger) -> bool

   Check if n is a Gaussian unit.

   :param n: An integer

   :returns: True if n is 1, -1, i or -i


.. py:function:: complex_factorise(n: arctans.numbers.GaussianInteger, istart: int = 0) -> list[arctans.numbers.GaussianInteger]

   Factorise a Gaussian integer into Gaussian primes.

   :param n: An integer

   :returns: A list of Gaussian primes