$n^4 + 4^n$ is a not a prime Prove that $n^4 + 4^n$ is not a prime for all $n > 1$ and $n \in \mathbb{N}$. This question appeared in the undergrad entrance exam of the Indian Statistical institute. When $n$ is even ...--prophetes.ai

$n^4 + 4^n$ is a not a prime Prove that $n^4 + 4^n$ is not a prime for all $n > 1$ and $n \in \mathbb{N}$. This question appeared in the undergrad entrance exam of the Indian Statistical institute. When $n$ is even the proof is simple. For $n = 2m+1$ I am utterly stuck.

Let $n=2k+1$ with $k\geq 1$, then $$n^4+4^n=n^4+4 \cdot 4^{2k}=n^4+4\cdot (2^k)^4=(n^2+2\cdot 2^{2k}+2^{k+1}n)(n^2+2\cdot 2^{2k}-2^{k+1}n).$$ Thus $n^4+4^n$ can be factored into non-trivial factors, when $n$ is odd.