Prove that $2^{4n}+1$ cannot be a prime if $3|n$

Question

$2^{4n}+1$ cannot be a prime if $3|n$ and $n>0$

My Try: $$2^{12k}+1\equiv (-1)^{3k}+1 \equiv0\pmod{17}$$ So it divisible by $17$ for odd $k$. But how to complete the proof?

Answer 1

Hint: $$ 16^{3k} +1 = (16^k+1)(16^{2k}-16^k+1).$$