Pell's Equation/Examples

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Examples of Solutions of Pell's Equation

Pell's equation: $x^2 - 13 y^2 = 1$

$x^2 - 13 y^2 = 1$

has the smallest positive integral solution:

$x = 649$
$y = 180$


Pell's equation: $x^2 - 29 y^2 = 1$

$x^2 - 29 y^2 = 1$

has the smallest positive integral solution:

$x = 9801$
$y = 1820$


Pell's equation: $x^2 - 61 y^2 = 1$

$x^2 - 61 y^2 = 1$

has the smallest positive integral solution:

$x = 1 \, 766 \, 319 \, 049$
$y = 226 \, 153 \, 980$