I'm new to the finite field theory, however after doing some trivial search on primitive polynomials, It seems that there are no primitive polynomial of the form $$x^{2k}+x^{k}+1\quad k>1$$ and I'm trying to prove it. To do so, I showed that $x^{2kt}+x^{kt}+1$ has a factor of $x^{2t}+x^{t}+1$ if $3\nmid k$ and hence is reducible over $\mathbb{F}_2$ and therefore it is not primitive.
The proof will complete if I show that there are no primitive polynomial of the form $$x^{2\cdot3^n}+x^{3^n}+1 \pmod 2$$
I think it isn't possible to apply the method of checking all of the factors of $2^{2\cdot 3^n}-1$ to find the order of the zeros (see this question). What should I do?
Note: I checked the claim for $n=\{1,2\}$ and it is true.
I see no difficulty. Since $1+x^k+x^{2k}=\frac{x^{3k}-1}{x^k-1}$ (putting $k=3^n$), the order of $x$ is at most $3k$, so it cannot generate the whole multiplicative group, so the polynomial is not primitive.