# Pell’s equation and simple continued fractions

###### Theorem 1.

Let $d$ be a positive integer which is not a perfect square^{}, and let $\mathrm{(}x\mathrm{,}y\mathrm{)}$ be
a solution of ${x}^{\mathrm{2}}\mathrm{-}d\mathit{}{y}^{\mathrm{2}}\mathrm{=}\mathrm{1}$. Then $\frac{x}{y}$ is a convergent^{} in the simple
continued fraction^{} expansion of $\sqrt{d}$.

###### Proof.

Suppose we have a non-trivial solution $x,y$ of Pell’s equation, i.e. $y\ne 0$. Let $x,y$ both be positive integers. From

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

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

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

$$ |

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

Title | Pell’s equation and simple continued fractions |
---|---|

Canonical name | PellsEquationAndSimpleContinuedFractions |

Date of creation | 2013-03-22 13:21:04 |

Last modified on | 2013-03-22 13:21:04 |

Owner | Thomas Heye (1234) |

Last modified by | Thomas Heye (1234) |

Numerical id | 9 |

Author | Thomas Heye (1234) |

Entry type | Theorem |

Classification | msc 11D09 |