Existence of nontrivial solutions of Pell's equation

Statement

Let $d$ be a positive integer that is not a perfect square. Then Pell's equation $x^2 - d y^2 = 1$ has a solution in integers with $y \neq 0$ . In fact, it has infinitely many solutions in integers.

Remarks

One proof uses Dirichlet's approximation theorem to find infinitely many $x/y$ with $|x^2 - d y^2|$ bounded. Another proof uses the periodic continued fraction of $\sqrt{d}$ . The smallest solution can be very large. For $d = 61$ , it is $(1766319049, 226153980)$ .

History

Indian mathematicians, such as Brahmagupta, studied the equation many centuries before European mathematicians. Lagrange proved that it always has infinitely many solutions.

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Diophantine equation concept
Pell's equation definition
Existence of nontrivial solutions of Pell's equation statement

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Tag	`0EM6C9SZN5`
URI	`m2://0EM6C9SZN5`
Kind	statement, theorem
Writing	W0 stub
MSC	11D09
Status	This copy of the site has no build data. Run `m2 status 0EM6C9SZN5`.
Uses	Pell's equation

Existence of nontrivial solutions of Pell's equation

Statement

Remarks

History

Reading path

Where this leads

Further reading

What this page needs for W1

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