Find all integer solutions to x2 + y2 + z2 = 51. Use "without loss of generality."
The Attempt at a Solution
My informal proof attempt:
Let x, y, z be some integers such that x, y, z = (0 or 1 or 2 or 3) mod 4
Then x2, y2, y2 = (0 or 1) mod 4
So x2 + y2 + z2 = [ (0 or 1) + (0 or 1) + (0 or 1) ] mod 4
Since 51 = 3 (mod 4) = x2 + y2 + z2, then x, y, z = (1 or 3) mod 4
It is obvious that the solution is the permutations of ##\pm1, \pm1, \pm7##.
It's my first proof course and I'm a little shaky. Is my logic correct? I feel like I took a leap from "Since 51..." to the solution, is there a more formal way to write that? I'm also not sure how to use wlog here. Thanks.