Pell's Equation/Examples/2/-1

Theorem
Pell's Equation:
 * $x^2 - 2 y^2 = -1$

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.

Proof
From Continued Fraction Expansion of $\sqrt 2$:
 * $\sqrt 2 = \sqbrk {1, \sequence 2}$

The cycle is of length is $1$.

By the solution of Pell's Equation, the only solutions of $x^2 - 2 y^2 = -1$ are:
 * ${p_r}^2 - 2 {q_r}^2 = \paren {-1}^r$

for $r = 1, 2, 3, \ldots$

From Convergents to Continued Fraction Expansion of $\sqrt 2$:

from which the solutions are obtained by taking the convergents with odd indices.

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