#### Solving the diophantine equation ax^{2}-by^{2}=c, using the LMM method

Here a > 0, b > 0, c ≠ 0. Also let D = ab.

This program was written by the author after studying page 84 of the recent book *Quadratic Diophantine Equations*, Titu Andreescu, Dorin Andrica, Developments in Mathematics, Vol. 40, Springer 2015.

We make the transformation X = ax, Y = y and look for the fundamental solutions of the transformed equation X^{2} - Dy^{2} = ac. If there are none, then the original equation has no integer solutions. Otherwise each fundamental solution (α, β) for which α is divisible by a, gives rise to a solution family
x = (α/a)u + bβv, y = βu + α v,

where u and v are integers satisfying the equation u^{2} - Dv^{2} = 1.
The method works for ac with up to say 20 digits, due to the limitations of the program used to factor c.

The program is a BCMath version of BC function `aa1()`.

*Last modified 11th September 2015*

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