Show that $3^{2n+1}-4^{n+1}+6^n$ is never prime for natural n except 1. Show that $3^{2n+1}-4^{n+1}+6^n$ is never prime for natural n except 1. I tried factoring this expression but couldn't get very far. It is simple to show for even n but odd n was more difficult, at least for me.

You can factor it as $(3^n-2^n)(3^{n+1}+4*2^n)=3^{2n+1}-3*6^n+4*6^n-4*4^n$. Here we juggle between $(ab)^n=a^nb^n$.
Since we can factorise it, to have a prime we need one of these factors to be 1, which only happens when $n$ is one, i.e the first term is $(3-2)$ and the second is 17. Note that the second term can't be 1 as it's an addition of two positive quantities.
