.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/number_theory/primality/plot_03_prime_counting.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_number_theory_primality_plot_03_prime_counting.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_number_theory_primality_plot_03_prime_counting.py:


The prime-counting function and the prime number theorem
================================================================

Sieves all primes up to 100,000 and compares the prime-counting
function :math:`\pi(n)` against the prime number theorem's
approximation :math:`n/\ln n`.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import math

    from mathematicskit.number_theory import is_prime_miller_rabin, sieve_of_eratosthenes
    from mathematicskit.number_theory.visualizers.plots import plot_prime_counting








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Sieve and count
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-23

.. code-block:: Python


    primes = sieve_of_eratosthenes(100000)
    print(f"pi(100000) = {len(primes)} (prime number theorem estimate: {100000 / math.log(100000):.1f})")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    pi(100000) = 9592 (prime number theorem estimate: 8685.9)




.. GENERATED FROM PYTHON SOURCE LINES 24-26

Cross-check the largest sieved prime with Miller-Rabin
-----------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 26-30

.. code-block:: Python


    largest = int(primes[-1])
    print(f"largest prime found: {largest}, Miller-Rabin agrees: {is_prime_miller_rabin(largest)}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    largest prime found: 99991, Miller-Rabin agrees: True




.. GENERATED FROM PYTHON SOURCE LINES 31-33

Plot pi(n) vs. the prime number theorem approximation
-----------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 33-35

.. code-block:: Python


    plot_prime_counting(2000)



.. image-sg:: /api/gallery/number_theory/primality/images/sphx_glr_plot_03_prime_counting_001.png
   :alt: Prime-counting function
   :srcset: /api/gallery/number_theory/primality/images/sphx_glr_plot_03_prime_counting_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    <Axes: title={'center': 'Prime-counting function'}, xlabel='n', ylabel='number of primes <= n'>




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.029 seconds)


.. _sphx_glr_download_api_gallery_number_theory_primality_plot_03_prime_counting.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_03_prime_counting.ipynb <plot_03_prime_counting.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_03_prime_counting.py <plot_03_prime_counting.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_03_prime_counting.zip <plot_03_prime_counting.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_