.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/number_theory/totient/plot_01_euler_totient.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_totient_plot_01_euler_totient.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_number_theory_totient_plot_01_euler_totient.py:


Euler's totient function and Euler's theorem
===================================================

Euler's :math:`\varphi(n)` counts the integers in :math:`1, \dots, n`
coprime to :math:`n`. His 1763 theorem,
:math:`a^{\varphi(n)} \equiv 1 \pmod n` for :math:`\gcd(a, n) = 1`,
generalizes Fermat's little theorem (:math:`\varphi(p) = p - 1` for a
prime :math:`p`). This script checks the definition and the theorem and
plots :math:`\varphi(n)`, whose top edge :math:`n - 1` is traced by the
primes.

.. GENERATED FROM PYTHON SOURCE LINES 15-22

.. code-block:: Python

    import math

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.number_theory import euler_totient, fast_mod_pow, is_prime_trial_division








.. GENERATED FROM PYTHON SOURCE LINES 23-25

phi(n) counts the coprime residues
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-30

.. code-block:: Python


    for n in (9, 10, 12, 36, 97):
        coprime = [k for k in range(1, n + 1) if math.gcd(k, n) == 1]
        print(f"phi({n}) = {euler_totient(n):>2}  (count of coprime k <= n: {len(coprime)})")





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

 .. code-block:: none

    phi(9) =  6  (count of coprime k <= n: 6)
    phi(10) =  4  (count of coprime k <= n: 4)
    phi(12) =  4  (count of coprime k <= n: 4)
    phi(36) = 12  (count of coprime k <= n: 12)
    phi(97) = 96  (count of coprime k <= n: 96)




.. GENERATED FROM PYTHON SOURCE LINES 31-33

Euler's theorem: a^phi(n) = 1 (mod n)
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 33-38

.. code-block:: Python


    holds = all(fast_mod_pow(a, euler_totient(n), n) == 1 for n in range(2, 300) for a in range(1, n) if math.gcd(a, n) == 1)
    print("Euler's theorem holds for every n < 300 and every coprime a:", holds)
    print(f"e.g. 7^phi(40) = 7^{euler_totient(40)} = {fast_mod_pow(7, euler_totient(40), 40)} (mod 40)")





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

 .. code-block:: none

    Euler's theorem holds for every n < 300 and every coprime a: True
    e.g. 7^phi(40) = 7^16 = 1 (mod 40)




.. GENERATED FROM PYTHON SOURCE LINES 39-41

The totient function up to 1000
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 41-54

.. code-block:: Python


    ns = np.arange(1, 1001)
    phis = np.array([euler_totient(int(n)) for n in ns])
    is_p = np.array([is_prime_trial_division(int(n)) for n in ns])

    fig, ax = plt.subplots()
    ax.plot(ns[~is_p], phis[~is_p], ".", ms=2, color="tab:blue", label="composite n")
    ax.plot(ns[is_p], phis[is_p], ".", ms=2, color="tab:red", label="prime n: phi(n) = n - 1")
    ax.set_xlabel("n")
    ax.set_ylabel("phi(n)")
    ax.set_title("Euler's totient function")
    ax.legend()
    plt.show()



.. image-sg:: /api/gallery/number_theory/totient/images/sphx_glr_plot_01_euler_totient_001.png
   :alt: Euler's totient function
   :srcset: /api/gallery/number_theory/totient/images/sphx_glr_plot_01_euler_totient_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.047 seconds)


.. _sphx_glr_download_api_gallery_number_theory_totient_plot_01_euler_totient.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_euler_totient.ipynb <plot_01_euler_totient.ipynb>`

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

      :download:`Download Python source code: plot_01_euler_totient.py <plot_01_euler_totient.py>`

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

      :download:`Download zipped: plot_01_euler_totient.zip <plot_01_euler_totient.zip>`


.. only:: html

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

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