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

As the title asks: does there exist $N$ such that, for any prime $p$ larger than $N$, the expression $x^4 +y^4$ takes on all values in $\mathbb{Z}/p\mathbb{Z}$?

Partial results:
If $p=4k+3$, it easily works
if $p=4k+1$, if $g$ is a primitive root modulo $p$, and $A_i = \left\{ g^k : k \equiv i \pmod{4} \right\}$, then at least three of $A_i (i=0,1,2,3)$ must be expressed.