# continued fraction of $\sqrt{10k+3}$

$$\sqrt{10k+3}.$$

Where $k$ is a positive integer.

All the best,

• By that you mean $k = \sqrt{10 k + 3}$? If so, square both sides and use quadratic formula.
– Mark
Commented Sep 14, 2018 at 16:01
• No. I want to find continued fraction $\sqrt{10k+3}$ , where $k$ is positive integer.
– d.y
Commented Sep 14, 2018 at 16:03
• There is no general way to find this. It depends on the value of $k$. Commented Sep 14, 2018 at 16:22
• There are some patterns for which the continued fraction really does follow a consistent pattern, such as $\sqrt{k^2 - 1}$ or $\sqrt {k^2 + 1}.$ They have a $k^2$ in them. The ones you ask about have no pattern at all. Commented Sep 14, 2018 at 18:12
• Do all numerators need to be $1$? A simple continued fraction? Commented Sep 20, 2018 at 12:53

• (+1) What was the motivation behind consider $x+1$ equal to the expression? Commented Sep 20, 2018 at 17:11
• Although $k$ is a positive integer, the similarity to $\sqrt{3}=1+0.73205080\dots$ is undeniable. It's actually possible to add bigger values to $x$, but as the continued fraction appears from a recursive application, making the recursion $x\left(x+2\right)$ simple, i.e., tightly close, $x\left(x+2\right)+1$ is advisable. Commented Sep 20, 2018 at 17:50
• the similarity to $\sqrt{3}=1+0.73205080\cdots$ is undeniable- Can you explain this statement? Why they need to be similar? Commented Sep 20, 2018 at 18:53