# Homework Help: (Number theory) Sum of three squares solution proof

1. Jan 26, 2017

### Xizel

1. The problem statement, all variables and given/known data

Find all integer solutions to x2 + y2 + z2 = 51. Use "without loss of generality."

2. Relevant equations

3. 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.

2. Jan 26, 2017

### Staff: Mentor

3. Jan 26, 2017

### Xizel

If my logic is correct, then x, y, z can take on values of 1, 3, 5, 7. I gave it some thought and I'm not seeing it

4. Jan 26, 2017

### Staff: Mentor

$2\cdot 25 + 1$
The remark "use w.l.o.g." probably refers to the assumption $x \geq y \geq z \geq 0$ which you could make.