Pell Equation Solver

This calculator solves the Pell equation \[x^2 - ny^2=1 \] where $n$ is a nonsquare positive integer. The calculator firstly finds the regular continued fraction $[a_0;a_1,a_2,\ldots]$ of $\sqrt{n}$ using continued fractions calculator and then calculate the sequence of convergents $p_k/q_k$ until they satisfy the equation above. This is the fundamental solution $(x_1,y_1)$. As the Pell equation has many solutions all remaining solutions may be calculated from the relation \[ x_m+y_m\sqrt{n} = (x_1+y_1\sqrt{n})^m \] by expanding the right side, equating coefficients of $\sqrt{n}$ on both sides, and equating the other terms on both sides. For example, the fundamental solution of \[x^2-7y^2=1\] is $(x_1,y_1)=(8,3)$. We calculate the solution $(x_2,y_2)$ from \[ x_2+y_2\sqrt{7} = (8+3\sqrt{7})^2 = 8^2 + 48\sqrt{7}+ (3\sqrt{7})^2 = 127+48\sqrt{7} \] so we have $(x_2,y_2)=(127,48)$ as another solution.

Pell Equation Calculator

Adblocker detected! Please consider reading this notice.

This website is made possible by displaying online advertisements to its visitors. Please consider supporting us by disabling your ad blocker.

Or add comnuan.com to your ad blocking whitelist.

×