If $n\in\mathbb N$ and $4^n+2^n+1$ is prime, prove that there exists an $m\in\mathbb N\cup\{0\}$ such that $n=3^m$.

If $n\in\mathbb N$ and $4^n+2^n+1$ is prime, prove that there exists an $m\in\mathbb N\cup\{0\}$ such that $n=3^m$.

I.e. if $4^n+2^n+1$ is prime, prove that $n=3^m$, where $m\in\mathbb N\cup\{0\}$.

I don't know how I could solve this, I don't have any ideas. I know that $4^n+2^n+1$ is obviously odd. Assume $m=0$, i.e. $n=1$. Then $3$ is prime. So that doesn't disprove what we're trying to show.

So assume $m>0$. Then $4^n+2^n+1>3$, so $4^n+2^n+1$ is either $3k+1$ or $3k+2$, where $k\in\mathbb N$. So either $4^n+2^n$ or $4^n+2^n-1$ is divisible by 3. I think it's not something that could help out at solving this. Just a little observation. I don't have any ideas.

Solution 1:

Hint. For prime $p\not=3$ we have $$(x^2+x+1)\mid (x^{2p}+x^p+1)$$ Proof.

\begin{align}x^{2p}+x^p+1&=\frac{x^{3p}-1}{x^p-1}\\&=\frac{(x^3-1)\left(\sum\limits_{k=0}^{p-1}x^{3k}\right)}{x^p-1}\\&=\frac{(x^2+x+1)\left(\sum\limits_{k=0}^{p-1}x^{3k}\right)}{(x^p-1)/(x-1)}\end{align} Now it suffices to prove \begin{align}\frac{x^p-1}{x-1}\big|\sum_{k=0}^{p-1}x^{3k}\end{align} Since $x^p\equiv 1\pmod{x^p-1}$, we see $x^{np+j}\equiv x^j\pmod{\frac{x^p-1}{x-1}}$. Now note that since $3\not\mid p$, for $k=0,1,\cdots,p-1$ the sequence $3k$ form a complete residue system of $p$. So \begin{align}\sum_{k=0}^{p-1}x^{3k}&\equiv\sum_{j=0}^{p-1}x^j\pmod{\frac{x^p-1}{x-1}}\\ & \equiv\frac{x^p-1}{x-1}\equiv 0\pmod{\frac{x^p-1}{x-1}}\end{align}

Solution 2:

If $n$ is not $3^m$, then $n=3^mr$ for some integer $r$, $r\gt1$, $\gcd(r,3)=1$.

$$(2^n-1)(4^n+2^n+1)=2^{3n}-1=2^{3^{m+1}r}-1=(2^{3^{m+1}}-1)q$$ for some integer $q\gt1$. Now $$\gcd(2^a-1,2^b-1)=2^c-1$$ where $c=\gcd(a,b)$, so $$\gcd(2^n-1,2^{3^{m+1}}-1)=2^{3^m}-1$$ Hence, $$\gcd(2^{3^{m+1}}-1,4^n+2^n+1)\gt1$$ But $2^{3^{m+1}}-1\lt4^n+2^n+1$, so $$\gcd(2^{3^{m+1}}-1,4^n+2^n+1)\lt4^n+2^n+1$$ Therefore, $4^n+2^n+1$ can't be prime.

Solution 3:

The case $n=1$ is trivial. Let $n\ge2$

By contradiction assume that $n=3^km$ where $3\not|m$, $m\ge2$ and let $$P_q(x)=x^{2q}+x^q+1\in\mathbb Z[x]$$ so $$P_n(x)=(x^{3^k})^{2m}+(x^{3^k})^{m}+1=P_m(x^{3^k})$$ We have $$P_m(j)=P_m(j^2)=j^{2m}+j^m+1=\frac{j^{3m}-1}{j^m-1}=0$$ hence $P_1(x)|P_m(x)$ in $\mathbb Z[x]$ so $$P_1(2^{3^k})|P_m(2^{3^k})=P_n(2)=4^n+2^n+1$$ and $$1<P_1(2^{3^k})<4^n+2^n+1$$ so $4^n+2^n+1$ isn't prime. Contradiction.