Is every integer a mixed sum of three squares? Lagrange's four-square theorem states that every natural number can be represented as the sum of four integer squares $n = a^2 + b^2 + c^2 + d^2$.  
Question: Is every integer a mixed sum of three integer squares $n = \pm a^2\pm b^2 \pm c^2$ ?
Note that the signs are independently positive or negative, for example $28 = 36-9+1$.

You can write every number $n$ in the form $a^2+b^2-c^2$.  Just pick $a$ so that $n-a^2$ is odd and then solve
$$\begin{align}
b+c&=n-a^2\\
b-c&=1
\end{align}$$
for $b$ and $c$:
$$\begin{align}
b&={n-a^2+1\over2}\\
c&={n-a^2-1\over2}
\end{align}$$
