I made a mistake in the final expression: It should be 2^{17}+2-2^{10} = 130050
We have from the triangle inequality, |x-a|+|b-x| ≥ (b-a) for all x in [a,b]
Applying this
|x-2|+\left|2^{16}-x\right| \ge 2^{16}-2 \\ \\ \left|x-2^2\right| + \left|2^{15}-x\right|\ge 2^{15}-4 \\ \\ .\\ .\\ .\\ \left|x-2^8\right| + \left|2^{9}-x\right|\ge 2^{9}-2^8
Thus LHS \ge 2^{16}-2 + 2^{15}-4+...+2^{9}-2^8 = 130050
The minimum will occur at the intersection of the intervals
\left[2,2^{16}\right], \left[2^2, 2^{15}\right],...,\left[2^8, 2^{9}\right]
which is obviously the interval \left[2^8, 2^{9}\right]