#### Solving the diophantine equation x^{2}- Dy^{2}=N, D > 0 and not a perfect square

This finds the fundamental solutions, if any, of the diophantine equation x^{2}-Dy^{2}=N, D > 0, D not a perfect square. These are the solutions (a,b) in each equivalence class with least b ≥ 0, where we have a ≥ 0 if the class is ambiguous. If (x_{1},y_{1})is the least positive solution of Pell's equation, the solutions (x,y) in the class containing (a,b) are given by x+y√D = ±(a+b√D)(x_{1}+y_{1}√D)^{n}, n ℇ ℤ.

We use T. Nagell's bounds for the fundamental solutions, which are, with small changes, sufficient as well as being necessary conditions.

See T. Nagell's *Theory of Numbers*, pages 204-212, BC program and Fundamental solutions to generalized Pell equations by John Robertson.

Note that this method is only useful for small D and N. In general, the LMM (Lagrange-Matthews-Mollin) method is the one to use - see CALC for a C version.
Warning: The program may halt prematurely, without an error message, due to running time limits.

If the Nagell upper bound exceeds 10^{6}, the program terminates after calculation of the least solution of Pell's equation.

We also produce a sequence of non-negative solutions (a,b), one from each equivalence class, with minimal b, using an algorithm of Frattini. See papers from The Jahrbuch database for Frattini's papers of 1891-2.

*Last modified 2nd September 2014*

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