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Here is a link to Wikipedia's entry on Continued fraction.

I found the following claim about Pell's equation:

Pell's equation

Continued fractions play an essential role in the solution of Pell's equation. For example, for positive integers $p$ and $q,$ and non-square $n,$ it is true that $p^2 − nq^2 = \pm 1\;$ if and only if $\;\dfrac pq$ is a convergent of the regular continued fraction for $\sqrt n$.

Question:

Isn't this claim false? For example, $\dfrac{39}{5}$ is a convergent of the regular continued fraction of $\sqrt{61};$ however, $39^2-5^2\cdot 61=-4\ne\pm1$.

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    Indeed, the claim is too strong. In the article on [Pell's Equation](https://en.wikipedia.org/wiki/Pell%27s_equation) they correctly say that the "fundamental solution" to Pell is a convergent, but it probably won't be the first one. In that article they analyze $n=7$ and point out that the first three convergents to $\sqrt 7$ do not work but the fourth does. – 2017-01-17
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    @lulu Is there an easy proof that the first convergent solving the Pell-equation is the fundamental solution ? – 2017-01-17
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    As I recall, it isn't too bad (but it has been a while). [this article](https://crypto.stanford.edu/pbc/notes/contfrac/pell.html) appears to sum it up efficiently. – 2017-01-17

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