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<h2><a name="IntegralPoints"></a>
IntegralPoints
</h2>

Computes all integral points on an elliptic curve in normal form.

<p><p>

<h3>
Syntax:
</h3>
<pre>L := IntegralPoints(k);
L := IntegralPoints(a,b);</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td></td>
</tr>
<tr>
<td>integer</td>
<td><pre>  k  </pre></td>
<td></td>
</tr>
<tr>
<td>integer</td>
<td><pre>  a  </pre></td>
<td></td>
</tr>
<tr>
<td>integer</td>
<td><pre>  b  </pre></td>
<td></td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
The <tt> IntegralPoints</tt> function computes all integral
points on an elliptic curve in normal form over the rational integers.
The result is returned as a list of all x-y pairs solving
the equation.\bigskip

<tt> IntegralPoints(k)</tt>
computes all integral points of Mordell's equation y^2 = x^3
+ k, where k is a rational integer. In this case the {\tt
IntegralPoints} function uses the algorithm described in
Wi3. \medskip

<tt> IntegralPoints(a,b)</tt>
computes all integral points of an elliptic curve
in short Weierstra\ssform y^2 = x^3 + ax + b with
rational integers a,b. The computation of all integral
points is reduced to another problem in which finitely many
associated quartic Thue are solved Stroe1,TzWe1. \medskip

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<p>
<hr>
<p>
<h3>
Example:
</h3>
Compute all integral points of y^2 = x^3 + 17. 
<pre>

kash> IntegralPoints(17);
> [ [ -2, -3 ], [ -2, 3 ], [ -1, -4 ], [ -1, 4 ], [ 2, -5 ], [ 2, 5 ], 
  [ 4, -9 ], [ 4, 9 ], [ 8, -23 ], [ 8, 23 ], [ 43, -282 ], [ 43, 282 ], 
  [ 52, -375 ], [ 52, 375 ], [ 5234, -378661 ], [ 5234, 378661 ] ]

</pre>

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