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Math Help - Need Help with proof

  1. #1
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    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?
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  2. #2
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    Quote Originally Posted by dbinnerley View Post
    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)
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