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I'm trying to practice finding the fundamental solution of Pell Equations, but have not found efficient ways of finding fundamental solutions. However, it seems that using continued fractions would provide an effective way of finding these solutions, according to the link. To better understand how to actually use continued fractions through examples (with numbers),

How would you find the fundamental solution of $x^2 - 95y^2 = 1$ and of $x^2 - 74y^2 = 1$ using continued fractions?

Compact
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    You only need the convergents corresponging to the continued fractions of $\sqrt{95}$ and $\sqrt{74}$ and determine the first satisfying the Pell-equation. – Peter Jun 07 '18 at 15:30
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    @DietrichBurde, the current question asks for an explicit computation and so is not a duplicate. I'd like to see it reopened. The answers in that question are not directly usable here. – lhf Jun 07 '18 at 15:37
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    Do you know how to compute the continued fraction for $\sqrt{95},$ for example? – saulspatz Jun 07 '18 at 15:40
  • Take the period of the continued fraction of $\sqrt{n}$, let's call it $h$. Then, compute $c_{h-1}$ where $c_m:=p_m/q_m$ denotes the $m^{th}$ partial convergent. The pair $(p_{h-1},q_{h-1})$ is your fundamental solution. If you follow this method correctly; you should get $(39,4)$ and $(3699, 430)$ as the fundamental solutions to your questions. – thesmallprint Jun 07 '18 at 15:52
  • A very good explanation, and a step-by-step example, can be found in section 6.2.2 of http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html#section6.1 – saulspatz Jun 07 '18 at 15:57
  • @lhf I added an answer for 29, 95, 74 at https://math.stackexchange.com/questions/1045127/how-to-find-a-fundamental-solution-to-pells-equation/2811551#2811551 – Will Jagy Jun 07 '18 at 16:20
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    @saulspatz I added an answer including 95, 74 at https://math.stackexchange.com/questions/1045127/how-to-find-a-fundamental-solution-to-pells-equation/2811551#2811551 where I had previously used a very general method. – Will Jagy Jun 07 '18 at 16:22
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    @WillJagy Thank you (+1). – Dietrich Burde Jun 07 '18 at 18:03

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