Solving the Pell equation

We illustrate recent developments in computational number theory by studying their implications for solving the Pell equation. We shall see that, if the solutions to the Pell equation are properly represented, the traditional continued fraction method for solving the equation can be significantly accelerated. The most promising method depends on the use of smooth numbers. As with many algorithms depending on smooth numbers, its run time can presently only conjecturally be established; giving a rigorous analysis is one of the many open problems surrounding the Pell equation.

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