Continued Fractions and their application to solving Pell's equations

Continued Fractions and their application to

solving Pell¡¯s equations

Number Theory: Fall 2009

Peter Khoury

University of Massachusetts-Boston, MA

Gerard D.Ko?

University of Massachusetts-Boston, MA

Number Theory: Fall 2009

University of Massachusetts Boston

Outline

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Pell¡¯s Equation and History

Problems Involving Pell¡¯s Equation

Continued Fractions

Solving Pell¡¯s Equations

Number Theory: Fall 2009

University of Massachusetts Boston

Pell¡¯s Equation and History

Pell¡¯s Equation

The quadratic Diophantine equation of the form

x 2 ? dy 2 = ¡À1

where d is a positive square free integer is called a Pell¡¯s equation.

Number Theory: Fall 2009

University of Massachusetts Boston

Pell¡¯s Equation and History

Pell¡¯s Equation

The quadratic Diophantine equation of the form

x 2 ? dy 2 = ¡À1

where d is a positive square free integer is called a Pell¡¯s equation.

In this presentation, we focus separately on the equations of the

form

x 2 ? dy 2 = 1 and x 2 ? dy 2 = ?1.

Number Theory: Fall 2009

University of Massachusetts Boston

Pell¡¯s Equation and History

Pell¡¯s Equation

The quadratic Diophantine equation of the form

x 2 ? dy 2 = ¡À1

where d is a positive square free integer is called a Pell¡¯s equation.

In this presentation, we focus separately on the equations of the

form

x 2 ? dy 2 = 1 and x 2 ? dy 2 = ?1.

The name Pell¡¯s equation comes from Euler who in a letter to

Goldbach confused the name of William Brouncker, the ?rst

mathematician who gave an algorithm to solve the equation, with

that of the English mathematician John Pell(1 March 1611-12

December 1685).

Number Theory: Fall 2009

University of Massachusetts Boston

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