.. _sphx_glr_api_gallery_number_theory_diophantine:

Diophantine equations
-----------------------

Linear and Pell Diophantine equation solvers.


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


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

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Diophantus&#x27;s Arithmetica asks for whole-number solutions of equations with more unknowns than constraints. For the linear case ax + by = c, integer solutions exist exactly when \gcd(a, b) divides c, and then they form one infinite family spaced evenly along the line. This script solves 12x + 18y = 30, shows that 12x + 18y = 31 has no solution, and plots the integer points on the line.">

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  .. image:: /api/gallery/number_theory/diophantine/images/thumb/sphx_glr_plot_01_linear_diophantine_thumb.png
    :alt:

  :doc:`/api/gallery/number_theory/diophantine/plot_01_linear_diophantine`

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      <div class="sphx-glr-thumbnail-title">Diophantus and the linear equation ax + by = c</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Fermat&#x27;s 1657 challenge: solve x^2 - Dy^2 = 1 in integers. The continued fraction of \sqrt D always yields the fundamental solution (Lagrange, 1768), but its size is wildly erratic -- for D = 61 it is x = 1766319049, while D = 60 needs only x = 31. This script reproduces Fermat&#x27;s D = 61 case, generates further solutions from the fundamental one, and plots the fundamental x for every non-square D \le 120.">

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  .. image:: /api/gallery/number_theory/diophantine/images/thumb/sphx_glr_plot_02_pell_equation_thumb.png
    :alt:

  :doc:`/api/gallery/number_theory/diophantine/plot_02_pell_equation`

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      <div class="sphx-glr-thumbnail-title">Pell's equation x^2 - Dy^2 = 1</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/number_theory/diophantine/plot_01_linear_diophantine
   /api/gallery/number_theory/diophantine/plot_02_pell_equation