Pell Equation Solver

This calculator solves the Pell equation $x^{2} - n y^{2} = 1$ where $n$ is a nonsquare positive integer. The calculator firstly finds the regular continued fraction $[a_{0}; a_{1}, a_{2}, \dots]$ 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 $x^{2} - 7 y^{2} = 1$ 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