Algorithm for an elliptic equation

[Rafajafar]

Hello,
I'm trying to create my own version of the Sieve of Atkin for my Algorithm class final project, but ran into a wall. I want to be able to create a method of algorithmically finding the number of integer coordinate pair solutions such that x > 0 and y > 0 for the following equations:

4x^2 + y^2 = n.
3x^2 + y^2 = n.
3x^2 - y^2 = n.

For a set of particular n determined earlier in the program. This is NOT a set of related equations. Each equation is separate from each other as different cases.

Now, I know how to do this analytically, but telling a computer to do this without using the brute force method of checking each and every number combination (which slows down the program by the order of N^2) is posing a problem.

I was wondering if perhaps anyone here knows of a technique in linear algebra that could speed this process up? Perhaps some matrix algebra manipulation I could apply to each equation to find the number of coordinate pair solutions faster?

Please, any help would be much appreciated.

 

[fresh_42]

This is a rather complicated question which is dealt with in algebraic geometry, subsection elliptic curves. I don't know about specific algorithms, but I would assume that there are some. Probably not linear in time, but maybe in $O(n\log n)$ or similar. So the size of $n$ determines whether it is worth searching and implementing. There is no short answer which you might have looked for.