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:

  N = 8*M + 3 - k^2
  i^2 + j^2 = N
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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.