Solution of Pell's Equation is a Convergent

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Theorem

Let $x = a, y = b$ be a positive solution to Pell's Equation $x^2 - n y^2 = 1$.


Then $\dfrac a b$ is a convergent of $\sqrt n$.


Proof

Let $a^2 - n b^2 = 1$.

Then we have:

$\left({a - b \sqrt n}\right) \left({a + b \sqrt n}\right) = 1$.

So:

$a - b \sqrt n = \dfrac 1 {a + b \sqrt n} > 0$

and so $a > b \sqrt n$.

Therefore:

\(\displaystyle \left\vert{\sqrt n - \frac a b}\right\vert\) \(=\) \(\displaystyle \frac {a - b \sqrt n} b\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b \left({a + b \sqrt n}\right)}\)
\(\displaystyle \) \(<\) \(\displaystyle \frac 1 {b \left({b \sqrt n + b \sqrt n}\right)}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 b^2 \sqrt n}\)
\(\displaystyle \) \(<\) \(\displaystyle \frac 1 {2 b^2}\)


The result follows from Condition for Rational to be a Convergent.

$\blacksquare$