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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.

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# Need help - Equation -> Algorithm for an elliptic equation

Can you offer guidance or do you also need help?

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