Originally Posted by

**kingwinner** **a) Prove that if $\displaystyle n = x^2 + y^2 + z^2$ and 4|n, then x, y, and z are even.**

**b) Prove that if n is of the form $\displaystyle 4^m(8k+7)$, then n is NOT the sum of three squares.**

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How can we prove part a?

For part b, this is what I've got so far.

Suppose n is a sum of three squares $\displaystyle n = x^2 + y^2 + z^2$ (aim for a contradiction). Assuming the result of part a, since 4|$\displaystyle 4^m(8k+7)$, x,y, and z must be even, so x/2, y/2, z/2 are integers.

=> $\displaystyle n/4 = (x/2)^2 + (y/2)^2 + (z/2)^2$, so n/4 is also a sum of three squares

How to finish the proof from here?

Any help is appreciated!

[note: also under discussion in math links forum]