Another way to write this is
\(x=(x^2-y)^{2016!} \)
x and y must both be integers
If x=0 and y=0 then 0=0 that works
If \(x=1\;\;and\;\;x^2-y=\pm1\)
that would work to if they are both integers that is
1-y=1
y=0
so
if x=1 and y=0 that works too
1-y=-1
2=y
so
if x=1 and y=2 that will work
and that is it 
so the 3 answers are (0,0) (1,0) and (1,2) 
So x+y = 0 or 1 or 3
Maybe another mathematician can do it a different / better way ??