## 3. Repeating sequence of continued fraction

### Continued fraction expansion and the solutions of Pell equation

Computing the solutions of Pell equation for n less than 100.

- integer part, length of fraction part,

fraction part for continued fraction expansion of sqrt(n)
- solutions of Pell equation
- x
^{2} - my^{2}

**Results of computation are here.**

#### Known results

the pattern of fraction part of continued fraction of sqrt(n) is always palindromic as

a_{1}, a_{2}, ...... , a_{i}, ...... , a_{2}, a_{1}, b

last "b" is twice of integer part

- right part of Pell equation's value is (-1)
^{k} where k is the length of fraction part

#### Remarkable patterns

- sqrt(n
^{2}+1) = n [2n]
- sqrt(n
^{2}+2) = n [n, 2n]

#### Excel work sheet for solving the Pell equation

E-mail : kc2h-msm@asahi-net.or.jpHisanori Mishima