Pell’s equation and simple continued fractions

Theorem 1.

Let d be a positive integer which is not a perfect squareMathworldPlanetmath, and let (x,y) be a solution of x2-dy2=1. Then xy is a convergentMathworldPlanetmathPlanetmath in the simple continued fractionMathworldPlanetmath expansion of d.


Suppose we have a non-trivial solution x,y of Pell’s equation, i.e. y0. Let x,y both be positive integers. From


we see that (xy)2>d, hence xy>d. So we get

|xy-d|=1y2(xy+d) <1y2(2d)

This implies that xy is a convergent of the continued fractionDlmfMathworld of 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