How to get all solutions for a negative Pell equation? For example, the equation $x^2 - 2 y^2 = -1$ has two solutions - $(7, 5)$ and $(41, 29)$, and the $(7, 5)$ is the fundamental one, right? How to get the $(41, 29)$ solution from the fundamental one?


The fundamental solution of the equation $x^2-2y^2=-1$ is $(1,1)$. We get all positive solutions by taking odd powers of $1+\sqrt{2}$. The positive solutions are $(x_n,y_n)$, where $x_n+y_n\sqrt{2}=(1+\sqrt{2})^{2n-1}$.

One can alternately obtain a recurrence for the solutions. If $(x_n,y_n)$ is a positive solution, then the "next" solution $(x_{n+1},y_{n+1})$ is given by $$x_{n+1}=3x_n+4y_n,\qquad y_{n+1}=2x_n+3y_n.$$

Note that your solution $(7,5)$ is the case $n=2$, and $(41,29)$ is the case $n=3$.

Similarly, the positive solutions of the equation $x^2-dy^2=-1$ (if they exist) are obtained by taking the odd powers of the fundamental solution.

  • $\begingroup$ Andre - thank you for the perfect answer $\endgroup$ – HEKTO Oct 19 '13 at 17:35
  • $\begingroup$ You are welcome. There is, as you can imagine, a lot of theory, though the characterization of the $d$ for which there is a solution is not completely satisfactory. The fundamental solution, when it exists, can be found by looking at the continued fraction expansion of $\sqrt{d}$. $\endgroup$ – André Nicolas Oct 19 '13 at 18:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.