I think this problem is from an old math olympiad, but not sure. The problem is:
Find all integer solutions to
\begin{equation}x^2+y^2+z^2=2^{11}.\end{equation}
I know that this can be describe as "find all coordinates with integer values inside (or on) a sphere with radius $\sqrt{2^{11}}=\sqrt{2048}\approx45.25$. However, I don't know how to arrive at the solution. A solution or any hints would be highly appreciated. I also know that people have studied geometry/integer problems like these, for instance Gauss circle problem but that's 2D (circles) not 3D (spheres).
I have found solutions using some online diophantine equation solver to be the following (I added the plus-minuses):
\begin{align} x1 &= 0, & y1 &= \pm32, & z1 &= \pm32 \\ x2 &= \pm32, & y2& = 0, & z2 &= \pm32 \\ x3 &= \pm32, & y3& = \pm32, & z3 &= 0 \end{align}
Hope you understand me!
if $$ x^2 + y^2 + z^2 \equiv 0 \pmod 4 \; , \; $$ then all three of $x,y,z$ are even. In your case, this observation is repeated a few times. Put another way, if $$ x^2 + y^2 + z^2 = 4n \; , \; $$ then $$ \left( \frac{x}{2 } \right)^2 +\left( \frac{y}{2 } \right)^2 +\left( \frac{z}{2 } \right)^2 = n$$ and so on
Since $$ 2^{11} = 4^5 \cdot 2,$$ you wind up solving $u^2 + v^2 + w^2 = 2$ (twelve answers) and multiplying all three variables by $32$