Is there an obvious reason why $4^n+n^4$ cannot be prime for $n\ge 2$? I searched a prime of the form $4^n+n^4$ with $n\ge 2$ and did not find one with
 $n\le 12\ 000$. 


*

*If $n$ is even, then $4^n+n^4$ is even, so it cannot be prime.

*If $n$ is odd and not divisible by $5$ , then $4^n+n^4\equiv (-1)+1\equiv 0 \pmod 5$.
So, $n$ must have the form $10k+5$.
For $n=35$ and $n=55$, the number $4^n+n^4$ splits into two primes
with almost the same size.
So, is there an obvious reason (like algebraic factors) that there is no prime I am looking for ?

An alternative to @labbhattacharjee’s exemplary response (for the case of odd $n$ only):
(1) $X^4 + 4=(X^2-2X+2)(X^2+2X+2)$; (2) $X^4+4a^4=(X^2-2aX+2a^2)(X^2+2aX+2a^2)$; (3) $X^4+4^{2k+1}=X^4+4\cdot2^{4k}$.
