<s>[INST] for what values of $n$ ,$n^4+4$ is composite number? for what values of $n$ ,$n^4+4$ is composite number?
$n^4+4=a.b\Rightarrow (n^2-2n+2)(n^2+2n+2)=ab$
what to conclude next?
Thank you
 [/INST] Let's set the factors equal to $1$: $$n^2 - 2n + 2 = 1 \implies n^2 - 2n + 1 = 0 \implies n = 1$$
Similarly, $$n^2 + 2n + 2 = 1 \implies n^2 + 2n + 1 = 0 \implies n = -1$$
We can check and see that when $n = \pm 1$, the value of $n^4 + 4$ is indeed prime. Hence, for any other $n$, namely $n \neq \pm 1$, this will be composite.
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