Pell's equation

The fundamental solution to the Diophantine equation $x^2-\mathrm{}{y}^2=1$ is given by $x=\mathrm{}$ and $y=\mathrm{}$.

Try entering the following equalities in your Javascript console:

Note that $\frac{x}{y}$ is a fractional approximation of $\sqrt{N}$ with $$\left|\frac{x}{y}-\sqrt{N}\right|\le \frac{1}{y\left(x+\lfloor \sqrt{N}\rfloor y\right)}=\frac{1}{\mathrm{}}\le {10}^{-\mathrm{}}$$

The continued fraction algorithm provides the best rational approximations of $\sqrt{N}$: $$\sqrt{\mathrm{}}=$$

In the previous formula, the same pattern is repeated indefinitely and the initial sequence is (incidentally, note that the first and last elements satisfy the equality $\mathrm{}=2\times \mathrm{}$ while the other elements form a palindrome). The values of $x$ and $y$ above were actually obtained by truncating the continued fraction to $$=\frac{\mathrm{}}{\mathrm{}}=\frac{x}{y}$$

Alternatively, these values can be obtained by the Chrakravala method:

where the following rules are used:

• The integer square root of $N$ is defined as $\lfloor \sqrt{N}\rfloor$.
• Given integers $x_1,y_1,k_1,\lambda$ scaling down by $\lambda$ is done by ${{{{{(\lambda x_1)}^2 - N {(\lambda y_1)}^2}} = {\lambda^2k_1}}} \implies {x_1^2 - Ny_1^2 = k_1}$.
• Given integers $x_1,y_1,k_1,x_2,y_2,k_2$ the samāsa operation is applying Brahmagupta's identity: $$\left\{ \begin{gathered} {x_1^2 - Ny_1^2 = k_1} \\ {x_2^2 - Ny_2^2 = k_2} \end{gathered} \implies {{\left( x_1x_2 + {Ny_1y_2} \right)^2 - N\left(x_1y_2+x_2y_1\right)^2} = {k_1k_2} } \right.$$
• In particular, given $x_1,y_1,k_1$, Bhaskara's minimization is to take ${x_2 = m}, {y_2 = 1}, {k_2 = m^2 - N}$ where $m > 0$ is chosen such that ${\frac{x_1+{my_1}}{k_1}} \in \mathbb Z$ and $\left|m^2 - N \right|$ is minimal.