When Is \( n^4 + 4 \) a Prime Number?

[Greg]

Here is this week's POTW:

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Given that $n$ is an integer greater than $0$, when is $n^4+4$ prime?

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Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!

 

[Greg]

Congratulations to Opalg, topsquark, kaliprasad, Olinguito and castor28 for their correct solutions. castor28's solution may be found below.

Spoiler

We have the polynomial identity:
$$
n^4 + 4 = (n^2 - 2n + 2)(n^2 + 2n + 2)
$$
Where both factors are integers if $n$ is an integer. The second factor is always greater than $1$, and the first one is greater than $1$ if $n>1$, in which case $n^4+4$ is composite. If $n=1$, $n^4+4=5$ is prime.

Therefore, $n^4+4$ is prime if and only if $n=1$.