Quick question regarding screening out "a few factors which would make a solution impossible": Didn't Gauss prove Fermat's polygonal theorem for triangle numbers, showing that every positive integer can be represented as a sum of at most three triangle numbers? If so, how can there be cases for which a solution is impossible? (Or are you saying that the method you're using only works for some cases?)
I'm talking about an intermediate step. I'm looking for three odd squares that add to the number 8*M+3. I pick the first number, k, by brute force working down from the square root. So then I have to solve:
There are choices for k that don't work. I want to eliminate them quickly and move on to the next k in the loop instead of spending time trying all combinations of i and j. Eventually I will find an answer, but that choice of k won't be part of it.
For example, if N is a multiple of an odd power of 3, the quadratic problem can't be solved in integers. So I can eliminate about 1/3 of the possible choices for k.