I made the following using induction:
If $n=1$ then the proposition is true: $5^{2^1} - 1=24$ is divisible by $2^{1+1} = 4$
Now I suppose that for a natural number $k$, $5^{2^k} - 1$ is divisible by $2^{k+1}$ is true. And I want to prove (using this) that the proposition is true for $n=k+1$ but I don't know how to do this.
I appreciate the help you give me.
Answer
Step $n+1$:
\begin{align}
5^{2^{n+1}}-1 &= 5^{2^n \times 2}-1 \\
&= (5^{2^n})^2-1 \\
&= (5^{2^n}-1)(5^{2^n}+1) \\
&=(k2^{n+1})(5^{2^n}+1) \end{align}
For the first factor above the hypothesis (step $n$) is used,
the second factor is even, say $2p$, since a power of $5$ is odd.
Combining:
$$(k2^{n+1})2p= (kp)2^{n+2}$$
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