# Math Help - Need Help with proof

1. ## Need Help with proof

I need to proove that if n=7(mod 8 ), then n does not equal x^2+y^2+z^2, for any x,y,z, integers. (the first equals sign is supposed the be a modulo symbol)

I've reduced the problem to finding that n=8s+7, for any s, but how do I prove that this does not equal the second statement above, or am i going in the right direction?

2. Originally Posted by dbinnerley
I need to proove that if n=7(mod 8 ), then n does not equal x^2+y^2+z^2, for any x,y,z, integers. (the first equals sign is supposed the be a modulo symbol)

I've reduced the problem to finding that n=8s+7, for any s, but how do I prove that this does not equal the second statement above, or am i going in the right direction?
Consider $0^2\equiv0\ (\text{mod}\ 8)$
$1^2\equiv1\ (\text{mod}\ 8)$
$2^2\equiv4\ (\text{mod}\ 8)$
$3^2\equiv1\ (\text{mod}\ 8)$
$4^2\equiv0\ (\text{mod}\ 8)$
$5^2\equiv1\ (\text{mod}\ 8)$
$6^2\equiv4\ (\text{mod}\ 8)$
$7^2\equiv1\ (\text{mod}\ 8)$

All are in {0,1,4}. Since 0 has no effect on the sum, and because you can eliminate even sums, there are very few cases to test

1
1+4
1+4+4
1+1+1

none of which is congruent to 7 (mod 8)