Pell’s equation and simple continued fractions
Theorem 1.
Let be a positive integer which is not a perfect square, and let be
a solution of . Then is a convergent
in the simple
continued fraction
expansion of .
Proof.
Suppose we have a non-trivial solution of Pell’s equation, i.e. . Let both be positive integers. From
we see that , hence . So we get
This implies that is a convergent of the continued fraction of
.
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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 |