Triangular and prime numbers
=============================

#. Save this function and the next one into a file named
   ``triangulars.py``.  Write the function :py:func:`is_triangular`
   that given an integer, ``n``, checks whether or not it is a
   triangular number. An integer, ``n``, is said to be a triangular
   number if it exists a base, ``b``, such that :math:`\sum_{i=1}^b i
   = n`. The function must return an integer corresponding to the base,
   ``b``, if ``n`` is triangular number or a zero otherwise. Examples:

   .. literalinclude:: is_triangular.txt
         :language: python3
         :lines: 3-10

   .. note::
        More tests are provided in file :download:`is_triangular.txt <is_triangular.txt>`

#. Write the function :py:func:`triangulars` that given an integer,
   ``maxn``, gets the triangular numbers from 1 to ``maxn`` (both
   included). The function must return a list of tuples, where each
   tuple represents a triangular number and is made up of two
   elements: the triangular number, ``n``, and the corresponding base,
   ``b``. Examples:

   .. literalinclude:: triangulars.txt
         :language: python3
         :lines: 3-

   .. note::
        More tests are provided in file :download:`triangulars.txt <triangulars.txt>`

	  
#. Save this function and the next ones into a file named
   ``prime.py``.  Write the function :py:func:`is_prime` that, given
   an integer, returns ``True`` if it is a prime number or ``False``
   otherwise. Examples:

   .. literalinclude:: is_prime.txt
         :language: python3
         :lines: 3-10

   .. note::
        More tests are provided in file :download:`is_prime.txt <is_prime.txt>`

   
#. Write an improved version of the previous function,
   :py:func:`improved_is_prime`, taking into account the two following
   properties: if ``n`` can't be divided by 2, it can't be divided by any
   even number, and all the divisors of ``n`` greater than
   :math:`int(\sqrt{n})` are redundant. See `Primality tests
   <https://en.wikipedia.org/wiki/Primality_test>`_ . Examples:

   .. literalinclude:: improved_is_prime.txt
         :language: python3
         :lines: 3-10

   .. note::
        More tests are provided in file :download:`improved_is_prime.txt <improved_is_prime.txt>`

#. Write the function :py:func:`iterations_is_prime` that, given a
   integer, returns ``True`` if it is prime or ``False``
   otherwise. This function must also count and return the number of
   performed iterations. Examples:

   .. literalinclude:: iterations_is_prime.txt
         :language: python3
         :lines: 3-10

   .. note::
        More tests are provided in file :download:`iterations_is_prime.txt <iterations_is_prime.txt>`

#. Write the improved version of the previous function,
   :py:func:`improved_iterations_is_prime`. Compare the number of
   iterations of this improved version with that of the not 
   improved one. Examples:

   .. literalinclude:: improved_iterations_is_prime.txt
         :language: python3
         :lines: 3-10

   .. note::
        More tests are provided in file :download:`improved_iterations_is_prime.txt <improved_iterations_is_prime.txt>`


#. Write the function :py:func:`prime` that, given an integer, ``maxn``,
   returns a list with the prime numbers less than or equal to
   ``maxn`` (included). This function must call function
   :py:func:`improved_is_prime`. Examples:

   .. literalinclude:: prime.txt
         :language: python3
         :lines: 3-

.. note::
        More tests are provided in file :download:`prime.txt <prime.txt>`

.. rubric:: Solution

A solution of these functions is provided in files :download:`triangulars.py <solutions/triangulars.py>` and :download:`prime.py <solutions/prime.py>`.