Pell's Equation/Examples/61
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Theorem
- $x^2 - 61 y^2 = 1$
has the smallest positive integral solution:
- $x = 1 \, 766 \, 319 \, 049$
- $y = 226 \, 153 \, 980$
Proof
From Continued Fraction Expansion of $\sqrt {61}$:
- $\sqrt {61} = \sqbrk {7, \sequence {1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14} }$
The cycle is of length is $11$.
By the solution of Pell's Equation, the only solutions of $x^2 - 61 y^2 = 1$ are:
- ${p_{11 r} }^2 - 61 {q_{11 r} }^2 = \paren {-1}^{11 r}$
for $r = 1, 2, 3, \ldots$
When $r = 1$ this gives:
- ${p_{11}}^2 - 61 {q_{11}}^2 = -1$
which is not the solution required.
When $r = 2$ this gives:
- ${p_{22} }^2 - 61 {q_{22} }^2 = 1$
From Convergents of Continued Fraction Expansion of $\sqrt {61}$:
- $p_{22} = 1 \, 766 \, 319 \, 049$
- $q_{22} = 226 \, 153 \, 980$
although on that page the numbering goes from $p_0$ to $p_{21}$, and $q_0$ to $q_{21}$.
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $61$