Periodic continued fraction

Definition

An infinite simple continued fraction $[a_0; a_1, a_2, \dots]$ is periodic if there are integers $N \ge 0$ and $k \ge 1$ with $a_{n+k} = a_n$ for all $n \ge N$ . It is purely periodic if this holds with $N = 0$ .

Remarks

One writes the repeating block with a bar, for example $\sqrt{2} = [1; \overline{2}]$ and $\sqrt{3} = [1; \overline{1, 2}]$ . By theorems of Euler and Lagrange, a real number has a periodic continued fraction if and only if it is a quadratic irrational. That is, it is an irrational root of a quadratic equation with integer coefficients. The period of $\sqrt{d}$ gives the fundamental solution of Pell's equation.

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Tag	`0XKJPFS2W0`
URI	`m2://0XKJPFS2W0`
Kind	definition
Writing	W0 stub
MSC	11A55
Defines	Periodic continued fraction
Uses	Simple continued fraction

Periodic continued fraction

Definition

Remarks

Reading path

Further reading

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