Re^2: Triangle Numbers Revisited

[tmoertel]

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?)

[tall_man]

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

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.

[tilly]

I hadn't known it, but you're right, Gauss proved that in his diary (which wasn't discovered until he had been dead 50 years).