Pell's Equation/Examples/61

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Theorem

Pell's Equation:

$x^2 - 61 y^2 = 1$

has the smallest positive integral solution:

$x = 1 \, 766 \, 319 \, 049$
$y = 226 \, 153 \, 980$


Proof

From Continued Fraction Expansion of $\sqrt {61}$:

$\sqrt {61} = \left[{7, \left\langle{1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14}\right\rangle}\right]$

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:

${p_{11 r} }^2 - 61 {q_{11 r} }^2 = \left({-1}\right)^{11 r}$

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