- #1

- 4

- 0

## Homework Statement

Find all integer solutions to x

^{2}+ y

^{2}+ z

^{2}= 51. Use "without loss of generality."

## Homework Equations

## 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 x

^{2}, y

^{2}, y

^{2}= (0 or 1) mod 4

So x

^{2}+ y

^{2}+ z

^{2}= [ (0 or 1) + (0 or 1) + (0 or 1) ] mod 4

Since 51 = 3 (mod 4) = x

^{2}+ y

^{2}+ z

^{2}, 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.