I hope it's not inappropriate asking this here. I stumbled upon this site recently while researching a Project Euler problem, now I figure I'd use it to ask about a recurring theme in these problems: quadratic Diophantine equations.
I've recently boiled down another Project Euler problem (I won't say which, it should be unrecognizable from the original problem and should probably be kept that way) to the following Diophantine equation:
I've been trying to use http://www.alpertron.com.ar/METHODS.HTM as a reference, but I seem to get lost in a sea of constants. And the steps that program on the bottom of that page takes don't seem to match what he says to do. I'd rather be able to understand the steps I'm taking anyway rather than just copying a method.
I'm interested in all positive integer values of n and I'm more or less given a solution exists with n=2. How would I go about finding the rest of the solutions? And how would I solve these kinds of equations in general? If that last part is too complex a question to be handled here, is there any other resource that might help? As far as my current level of math, I have a degree in engineering (and helped a math major with some courses I never took myself) and I've already worked on Project Euler problems involving Pell's equations and continued fraction expansions of square roots.