PlanetMath: Pell's equation and simple continued fractions

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Pell's equation and simple continued fractions

(Theorem)

Theorem 1 Let be a positive integer which is not a perfect square, and let be a solution of . Then $\frac{x}{y}$ is a convergent in the simple continued fraction expansion of $\sqrt{d}$ .

Proof. Suppose we have a non-trivial solution

of Pell's equation, i.e. $y \neq 0$ . Let

both be positive integers. From

$\displaystyle \left(\frac{x}{y}\right)^2 =d +\frac{1}{y^2}$

we see that $\left(\frac{x}{y}\right)^2 > d$ , hence $\frac{x}{y} > \sqrt{d}$ . So we get

$\displaystyle \left\vert \frac{x}{y} -\sqrt{d}\right\vert =\frac{1}{y^2\left(\frac{x}{y} +\sqrt{d}\right)}$		$\displaystyle < \frac{1}{y^2\left(2\sqrt{d}\right)}$
		$\displaystyle < \frac{1}{2y^2}.$

This implies that $\frac{x}{y}$ is a convergent of the continued fraction of $\sqrt{d}$ . $\qedsymbol$

"Pell's equation and simple continued fractions" is owned by Thomas Heye.

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Cross-references: continued fraction, implies, Pell's equation, simple continued fraction, convergent, solution, perfect square, integer, positive
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This is version 6 of Pell's equation and simple continued fractions, born on 2003-01-04, modified 2006-10-10.
Object id is 3870, canonical name is PellsEquationAndSimpleContinuedFractions.
Accessed 4100 times total.

Classification:

AMS MSC:

11D09 (Number theory :: Diophantine equations :: Quadratic and bilinear equations)

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