Pell's Equation/Examples/2

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Theorem

Pell's Equation:

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

has the positive integral solutions:

$\begin {array} {r|r} x & y \\ \hline 3 & 2 \\ 17 & 12 \\ 99 & 70 \\ 577 & 408 \\ 3363 & 2378 \\ \end {array}$

and so on.


Proof

From Continued Fraction Expansion of $\sqrt 2$:

$\sqrt 2 = \sqbrk {1, \sequence 2}$

The cycle is of length is $1$.

By the solution of Pell's Equation, the only solutions of $x^2 - 2 y^2 = -1$ are:

${p_r}^2 - 2 {q_r}^2 = \paren {-1}^r$

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

From Convergents to Continued Fraction Expansion of $\sqrt 2$:

The sequence of convergents to the continued fraction expansion of the square root of $2$ begins:

$\dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \dfrac {1393} {985}, \dfrac {3363} {2378}, \ldots$


from which the solutions are obtained by taking the convergents with even indices.

$\blacksquare$