.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/number_theory/quadratic_residues/plot_01_quadratic_reciprocity.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_quadratic_residues_plot_01_quadratic_reciprocity.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_number_theory_quadratic_residues_plot_01_quadratic_reciprocity.py:


The law of quadratic reciprocity
===================================================

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

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.number_theory import jacobi_symbol, legendre_symbol, sieve_of_eratosthenes, sqrt_mod








.. GENERATED FROM PYTHON SOURCE LINES 21-23

Tabulate (p/q) and check reciprocity
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 23-31

.. code-block:: Python


    primes = [int(p) for p in sieve_of_eratosthenes(100)[1:]]
    table = np.array([[legendre_symbol(p, q) if p != q else 0 for p in primes] for q in primes])
    sign = np.array([[-1 if p % 4 == 3 and q % 4 == 3 else 1 for p in primes] for q in primes])
    off_diagonal = ~np.eye(len(primes), dtype=bool)
    holds = bool(np.all((table * table.T == sign)[off_diagonal]))
    print("quadratic reciprocity holds for all pairs below 100:", holds)





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

 .. code-block:: none

    quadratic reciprocity holds for all pairs below 100: True




.. GENERATED FROM PYTHON SOURCE LINES 32-34

Reciprocity lets the Jacobi symbol be computed without factoring,
and Tonelli-Shanks then produces the square root itself.

.. GENERATED FROM PYTHON SOURCE LINES 34-48

.. code-block:: Python


    print(f"(1001 / 9907) = {jacobi_symbol(1001, 9907)}")
    x = sqrt_mod(10, 13)
    print(f"sqrt(10) mod 13 = {x}  (check: {x}^2 mod 13 = {x * x % 13})")

    fig, ax = plt.subplots(figsize=(6, 5))
    im = ax.imshow(table, cmap="coolwarm", vmin=-1, vmax=1)
    ax.set_xticks(range(len(primes)), primes, fontsize=6, rotation=90)
    ax.set_yticks(range(len(primes)), primes, fontsize=6)
    ax.set_xlabel("p")
    ax.set_ylabel("q")
    ax.set_title("Legendre symbol (p / q)")
    fig.colorbar(im, ax=ax, ticks=[-1, 0, 1])
    plt.show()



.. image-sg:: /api/gallery/number_theory/quadratic_residues/images/sphx_glr_plot_01_quadratic_reciprocity_001.png
   :alt: Legendre symbol (p / q)
   :srcset: /api/gallery/number_theory/quadratic_residues/images/sphx_glr_plot_01_quadratic_reciprocity_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    (1001 / 9907) = -1
    sqrt(10) mod 13 = 6  (check: 6^2 mod 13 = 10)





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

   **Total running time of the script:** (0 minutes 0.046 seconds)


.. _sphx_glr_download_api_gallery_number_theory_quadratic_residues_plot_01_quadratic_reciprocity.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_quadratic_reciprocity.ipynb <plot_01_quadratic_reciprocity.ipynb>`

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

      :download:`Download Python source code: plot_01_quadratic_reciprocity.py <plot_01_quadratic_reciprocity.py>`

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

      :download:`Download zipped: plot_01_quadratic_reciprocity.zip <plot_01_quadratic_reciprocity.zip>`


.. only:: html

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

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