Chapter 8 : Continued Fraction and Pell's Equation

(First : June 16, 1997; update : April 12, 2008) [Japanese]


Diophantine equation x2-ny2=±1 is called Pell equation.
This equation always has non-trivial solution ("trivial solution" is x=1, y=0.)
Sometimes x and y becomes large for some n. For example,

2275282 - 103 * 224192 = 1
88901822 - 109 * 8515252 = -1

In this chapter, first try to find the solutions for all n under 100 in primitive way,
and next, solve by using continued fraction expansion of sqrt(n).
And also show the Excel sheet to solve Pell equation.

In order to understand the difficulty of this problem, please try to solve the following questions.


  1. Solution by brute force method
  2. Solution by continued fraction expansion
  3. Repeating sequence of continued fraction


[1] Albert H. Beiler, "Recreations in the Theory of Numbers (Second Edition),
     The Queen of Mathematics Entertains", Dover (1966)

Chapter 7
Collatz's Conjecture
"Mathematician's Secret Room" Chapter 9
Amicable Numbers
Chapter 7 (Japanese) index (Japanese) Chapter 9 (Japanese)

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Hisanori Mishima