<s>[INST] How to prove $8^{2^n} - 5^{2^n}$ is divisible by $13$ where $n\in\mathbb{N}-\{0\}$ with induction? So the question is as above is shown:

How to prove $8^{2^n} - 5^{2^n}$ is divisible by $13$?

$n=1: 8^2 - 5^2 = 39$ is a multiple of $13$.
I.H.: Suppose $8^{2^n} - 5^{2^n}$ is divisible by $13$ is true $\forall n \in \mathbb{N}\setminus\{0\}$. 
I got stuck on this $n= m + 1$ case.
$8^{2^{m+1}} - 5^{2^{m+1}}= 8^{2^m\cdot 2}- 5^{2^m\cdot 2}$
Can somebody please help me?
 [/INST] To complete the proof with the following identity $ A^2 - B^2 = (A + B) (A-B) $. 
I get 
$\begin{align}\\
\underline {\text{n= m+1:}} \\
8^{2^n} -5^{2^n} & = 8^{2^{m+1}} -5^{2^{m+1}} \\
& = (8^{2^m} + 5^{2^m})(8^{2^m} -5^{2^m})\\
& = (8^{2^m} + 5^{2^m})\cdot \text{some multiple of 13}
\end{align}$
Thus concluding that the entire expression is divisible by 13.
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