.. _sphx_glr_api_gallery_number_theory_sums_of_squares:

Sums of squares
---------------

Fermat's two-squares theorem and Lagrange's four-square theorem.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Every prime p \equiv 1 \pmod 4 is a sum of two squares a^2 + b^2, and no prime p \equiv 3 \pmod 4 is. This script checks the theorem for every prime below 10,000 and plots each representation as a lattice point (a, b) on the circle of radius \sqrt p.">

.. only:: html

  .. image:: /api/gallery/number_theory/sums_of_squares/images/thumb/sphx_glr_plot_01_fermat_two_squares_thumb.png
    :alt:

  :doc:`/api/gallery/number_theory/sums_of_squares/plot_01_fermat_two_squares`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Fermat's two-squares theorem</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Every non-negative integer is a sum of four squares. This script finds a representation for every n &lt; 10{,}000 and plots the fewest squares each n actually needs: one for perfect squares, two by Fermat&#x27;s criterion, four exactly for n = 4^k(8m+7) (Legendre&#x27;s three-square theorem), and three otherwise.">

.. only:: html

  .. image:: /api/gallery/number_theory/sums_of_squares/images/thumb/sphx_glr_plot_02_lagrange_four_squares_thumb.png
    :alt:

  :doc:`/api/gallery/number_theory/sums_of_squares/plot_02_lagrange_four_squares`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Lagrange's four-square theorem</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/number_theory/sums_of_squares/plot_01_fermat_two_squares
   /api/gallery/number_theory/sums_of_squares/plot_02_lagrange_four_squares