# Need Help with proof

• Jul 13th 2010, 01:40 PM
dbinnerley
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?
• Jul 13th 2010, 01:50 PM
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Quote:

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 $\displaystyle 0^2\equiv0\ (\text{mod}\ 8)$
$\displaystyle 1^2\equiv1\ (\text{mod}\ 8)$
$\displaystyle 2^2\equiv4\ (\text{mod}\ 8)$
$\displaystyle 3^2\equiv1\ (\text{mod}\ 8)$
$\displaystyle 4^2\equiv0\ (\text{mod}\ 8)$
$\displaystyle 5^2\equiv1\ (\text{mod}\ 8)$
$\displaystyle 6^2\equiv4\ (\text{mod}\ 8)$
$\displaystyle 7^2\equiv1\ (\text{mod}\ 8)$

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)