If $n\ge 3$, $4^n \nmid 9^n-1$ Could anyone give me a hint to prove the following?
If $n\ge 3$, $4^n \nmid 9^n-1$

Hint :
Try to prove using  induction :
$1.$ $9^3 \not \equiv 1 \pmod {4^3}$
$2.$ suppose : $9^k \not \equiv 1 \pmod {4^k}$
$3.$ $9^k \not \equiv 1 \pmod {4^k} \Rightarrow 9^{k+1} \not \equiv 9 \pmod {4^k}$
So you have to prove :
$ 9^{k+1} \not \equiv 9 \pmod {4^k} \Rightarrow  9^{k+1} \not \equiv 1 \pmod {4^{k+1}}$
