immaterial due to the quadratic nature of the equation) which gives an infinite tower of solutions to the negative Pell's equation. Generalized Pell's equation The equation is called the generalized (or general) Pell's equation. The equation is the corresponding Pell's resolvent. A recursive algorithm was given by Lagrange in 1768 for solving the equation, reducing the problem to the case . Such solutions can be derived using the continued-fractions method as outlined above. If is a solution to and is a solution to then such that is a solution to , a principle named the multiplicative principle. The solution is called a Pell multiple of the solution . There exists a finite set of solutions to such that every solution is a Pell multiple of a solution from that set. In particular, if is the fundamental solution to , then each solution to the equation is a Pell multiple of a solution with and , where . If x and y are positive integer solutions to the Pell's equation with , then is a convergent to the continued fraction of . Solutions to the generalized Pell's equation are used for solving certain Diophantine equations and units of certain rings, and they arise in the study of SIC-POVMs in quantum information theory. The equation is similar to the resolvent in that if a minimal solution to can be found, then all solutions of the equation can be generated in a similar manner to the case . For certain , solutions to can be generated from those with , in that if then every third solution to has even, generating a solution to . Notes References Further reading External links Pell equation solver (n has no upper limit) Pell equation solver (n < 1010, can also return the solution to x2 − ny2 = ±1, ±2, ±3, and ±4) Diophantine equations Continued fractions