As the title, for any prime $p$ larger than some $N$,  can $x^4 +y^4$ express all numbers in $\mathbb{Z}_p $ ?  
I'm thinking about this problem for days, but I failed to solve it. Does anyone know whether this is true, or any partial results about it?   


Partial results  
If $p=4k+3$ , it easily works  
if $p=4k+1$, if we say $g$ is a primitive root of $p$ and $A_i = [g^k |k \equiv i(mod 4)]$ , then at least three of $A_i (i=0,1,2,3)$ must be expressed