Pell's Equation/Examples/13
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Theorem
- $x^2 - 13 y^2 = 1$
has the smallest positive integral solution:
- $x = 649$
- $y = 180$
Proof
From Continued Fraction Expansion of $\sqrt {13}$:
- $\sqrt {13} = \sqbrk {3, \sequence {1, 1, 1, 1, 6} }$
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 $q_9$.
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $13$