Pell's Equation/Examples/13

Theorem
Pell's Equation:
 * $x^2 - 13 y^2 = 1$

has the smallest positive integral solution:
 * $x = 649$
 * $y = 180$

Proof
From Continued Fraction Expansion of $\sqrt {13}$:
 * $\sqrt {13} = \sqbrk {3, \sequence {1, 1, 1, 1, 6} }$

The cycle is of length $5$.

By the solution of Pell's Equation, the only solutions of $x^2 - 13 y^2 = 1$ are:
 * ${p_{5 r} }^2 - 13 {q_{5 r} }^2 = \paren {-1}^{5 r}$

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

When $r = 1$ this gives:
 * ${p_5}^2 - 13 {q_5}^2 = -1$

which is not the solution required.

When $r = 2$ this gives:
 * ${p_{10} }^2 - 13 {q_{10} }^2 = 1$

From Convergents of Continued Fraction Expansion of $\sqrt {13}$:
 * $p_{10} = 649$
 * $q_{10} = 180$

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