Show that the number $2^{2^{2n+1}}+2^{2^{2n}}+1$ is not prime.

Question

question

Show that the number $2^{2^{2n+1}}+2^{2^{2n}}+1$ is not prime, for any nonzero natural number $n$.

my idea

$ 2^{2^{2n+1}}+2^{2^{2n}}+1 = 2^{2^{2n}}*5+1 $ i did this by giving common factor.

From here we can easily observe that the last digit of thiz equation is $1$ and also this equation has the form $k^2*5+1$. I tried using some modular arithmetic but i obteined nothing. I got stuck here and i dont knkw what to do forward.

Hope one of you can help me! Thank you!

Answer 1

$2^{2^{2n+1}} = 2^{[2 \cdot (2^{2n})]} = (2^{2^{2n}})^2$

So you end up with $X^2+X+1$ where $X=2^{2^{2n}}$. I guess you can start from there.
It's easy to see that, if $X \mod 3 = 1$, then $3 | X^2+X+1$, so I guess you need to move in that direction.

Answer 2

Both $2^{2n+1}$ and $2^{2n}$ are even, thus $2^{2^{2n+1}}\equiv (-1)^{2^{2n+1}}\equiv 1$ and $2^{2^{2n}}\equiv (-1)^{2^{2n}}\equiv 1$ modulo 3. It follows that $2^{2^{2n+1}}+2^{2^{2n}}+1$ is divisible by 3, and so it is not prime.