Pell's Equation/Examples/8

Theorem
Pell's Equation:
 * $x^2 - 8 y^2 = 1$

has the positive integral solutions:

and so on.

Proof
From Continued Fraction Expansion of $\sqrt 8$:
 * $\sqrt 8 = \sqbrk {2, \sequence {1, 4} }$

The cycle is of length $2$.

By the solution of Pell's Equation, the only solutions of $x^2 - 8 y^2 = 1$ are:
 * ${p_{2 r} }^2 - 8 {q_{2 r} }^2 = \paren {-1}^{2 r}$

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

When $r = 1$ this gives:
 * ${p_2}^2 - 8 {q_2}^2 = 1$

which is the solution required.

From Convergents of Continued Fraction Expansion of $\sqrt 8$:
 * $p_2 = 3$
 * $q_2 = 1$

although on that page the numbering goes from $p_0$ to $p_9$, and $q_0$ to $q_9$.

[[Category:8]]