Pell's Equation/Examples

Examples of Solutions of Pell's Equation

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

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

has the positive integral solutions:

$\begin {array} {r|r} x & y \\ \hline

3 & 2 \\ 17 & 12 \\ 99 & 70 \\ 577 & 408 \\ 3363 & 2378 \\ \end {array}$

and so on.

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

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

has the positive integral solutions:

$\begin {array} {r|r} x & y \\ \hline

1 & 1 \\ 7 & 5 \\ 41 & 29 \\ 239 & 169 \\ 1393 & 985 \\ \end {array}$

and so on.

Pell's equation: $x^2 - 8 y^2 = 1$

$x^2 - 8 y^2 = 1$

has the positive integral solutions:

\(\ds \tuple {x, y}\)

\(=\)

\(\ds \tuple {3, 1}\)

\(\ds \tuple {x, y}\)

\(=\)

\(\ds \tuple {17, 6}\)

\(\ds \tuple {x, y}\)

\(=\)

\(\ds \tuple {99, 35}\)

\(\ds \tuple {x, y}\)

\(=\)

\(\ds \tuple {577, 204}\)

\(\ds \tuple {x, y}\)

\(=\)

\(\ds \tuple {3363, 1189}\)

and so on.

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:

$x = 1 \, 766 \, 319 \, 049$

$y = 226 \, 153 \, 980$