<s>[INST] $n^4 + 4^n$ 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.
 [/INST] Firstly, I’ll show that we can factorise $x^4+4y^4$.
$$x^4+4y^4=x^4-4x^2y^2+4y^4-4x^2y^2= \left(x^2+2y^2\right)^2-\left(2xy\right)^2= \left(x^2+2xy+2y^2\right)\left(x^2-2xy+2y^2\right)$$
Back to the question, when $n=2m+1$ that $m$ is a positive integer, we can substitute $x=n$ and $y=2^m$ into the polynomial above, we’ll get
$$n^4+4\times\left(2^m\right)^4=n^4+4\times4^{2m}=n^4+4^{2m+1}=n^4+4^n=\left(n^2+2^{m+1}n+2^{2m+1}\right)\left(n^2-2^{m+1}n+2^{2m+1}\right)$$
Therefore, $n^4+4^n$ is not a prime for all odd number $n>1$
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