theorem   Let $$n$$ be a positive integer,  then $$n$$ can be expressed as the sum of three squares iff it is not of  the  form

$$4^a(8b+7)$$

for some $$a,b\in\Bbb Z,a,b\geq 0$$.

For a integer $$n\equiv3\pmod8$$, there exists three odd numbers $$x,y,z$$ such that

$$n=x^2+y^2+z^2.$$

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