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
• x² - my²

Results of computation are here.

Known results

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

a₁, a₂, ...... , a_i, ...... , a₂, a₁, 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(n²+1) = n [2n]
• sqrt(n²+2) = n [n, 2n]