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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 ?