.. 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_02_perfect_numbers.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_02_perfect_numbers.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_number_theory_totient_plot_02_perfect_numbers.py:


Perfect numbers and the divisor-sum function
===================================================

A perfect number equals the sum of its own proper divisors
(:math:`\sigma(n) = 2n`). This script finds the perfect numbers below
10,000 with
:func:`~mathematicskit.number_theory.systems.totient.divisor_sum` and
checks Euclid's form :math:`2^{p-1}(2^p - 1)` for each.

.. GENERATED FROM PYTHON SOURCE LINES 13-15

.. code-block:: Python

    from mathematicskit.number_theory import divisor_sum








.. GENERATED FROM PYTHON SOURCE LINES 16-18

Find perfect numbers up to 10,000
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 18-22

.. code-block:: Python


    perfect_numbers = [n for n in range(2, 10000) if divisor_sum(n) == 2 * n]
    print("perfect numbers below 10000:", perfect_numbers)





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

 .. code-block:: none

    perfect numbers below 10000: [6, 28, 496, 8128]




.. GENERATED FROM PYTHON SOURCE LINES 23-25

Each has Euclid's form 2^(p-1) (2^p - 1)
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-29

.. code-block:: Python


    for n in perfect_numbers:
        p = (n & -n).bit_length()  # 2^(p-1) is the largest power of 2 dividing n
        print(f"  {n} = 2^{p - 1} * (2^{p} - 1) -> {2 ** (p - 1) * (2**p - 1) == n}")




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

 .. code-block:: none

      6 = 2^1 * (2^2 - 1) -> True
      28 = 2^2 * (2^3 - 1) -> True
      496 = 2^4 * (2^5 - 1) -> True
      8128 = 2^6 * (2^7 - 1) -> True





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

   **Total running time of the script:** (0 minutes 0.014 seconds)


.. _sphx_glr_download_api_gallery_number_theory_totient_plot_02_perfect_numbers.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_perfect_numbers.ipynb <plot_02_perfect_numbers.ipynb>`

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

      :download:`Download Python source code: plot_02_perfect_numbers.py <plot_02_perfect_numbers.py>`

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

      :download:`Download zipped: plot_02_perfect_numbers.zip <plot_02_perfect_numbers.zip>`


.. only:: html

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

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