Pell's Equation/Examples
Jump to navigation
Jump to search
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:
- $x = 1 \, 766 \, 319 \, 049$
- $y = 226 \, 153 \, 980$