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

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The discussion centers on determining when the expression \( n^4 + 4 \) results in a prime number for integers \( n > 0 \). Participants explore various integer values and mathematical properties to identify conditions under which the expression yields a prime. Several users, including Opalg, topsquark, kaliprasad, Olinguito, and castor28, successfully provide correct solutions. Notably, castor28 shares a detailed solution that contributes to the understanding of the problem. The exploration highlights the relationship between the expression and prime number generation.
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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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Congratulations to Opalg, topsquark, kaliprasad, Olinguito and castor28 for their correct solutions. castor28's solution may be found below.

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 $n=1$.
 
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