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.