TITLE: $4^{2^n}+2^{2^n}+1$ is Divisible by $7$
QUESTION [9 upvotes]: I have one question.
How do I prove that $$4^{2^n}+2^{2^n}+1$$ is Divisible by $7$ ?
thanks in advances.

REPLY [2 votes]: Let $2^{2^n}=t_n$, then $t_n=t_{n-1}^2$. We want to show that $t^2_n+t_n+1$ is divisible by $7=4+2+1=t^2_0+t_0+1$. Now
$$\begin{aligned}t_n^2+t_n+1&=(t_n+1)^2-t_n\\
&=(t_{n-1}^2+1)^2-t_{n-1}^2\\
&=(t_{n-1}^2+t_{n-1}+1)(t_{n-1}^2-t_{n-1}+1).
\end{aligned}$$
The result follows by reduction/induction.