.. _sphx_glr_api_gallery_number_theory_quadratic_residues:

Quadratic residues
------------------

Legendre and Jacobi symbols, quadratic reciprocity, and modular square roots.


.. 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="For distinct odd primes p, q, \left(\frac{p}{q}\right)\left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}: whether p is a square mod q is tied to whether q is a square mod p. The plot tabulates the Legendre symbol for odd primes below 100; reflecting in the diagonal flips the sign exactly when both primes are 3 \bmod 4.">

.. only:: html

  .. image:: /api/gallery/number_theory/quadratic_residues/images/thumb/sphx_glr_plot_01_quadratic_reciprocity_thumb.png
    :alt:

  :doc:`/api/gallery/number_theory/quadratic_residues/plot_01_quadratic_reciprocity`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The law of quadratic reciprocity</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/number_theory/quadratic_residues/plot_01_quadratic_reciprocity