<s>[INST] computing the product $\prod_{n=1}^{2016} \frac{2^{2^{n-1}}+1}{2^{2^{n-1}}}$ how can i calculate the product: $\prod_{n=1}^{2016} \frac{2^{2^{n-1}}+1}{2^{2^{n-1}}}$?
I can see that in the denominator it's a geometric series, but in the numerator i can't see how to simplify.
Thanks.
 [/INST] Note that
\begin{align*}
\prod_{n=1}^{2016} \frac{2^{2^{n-1}}+1}{2^{2^{n-1}}}&
=\frac{(2^{2^{0}}-1)(2^{2^{0}}+1)\cdots (2^{2^{2015}}+1)}{(2^{2^{0}}-1)2^{2^{0}}\cdots 2^{2^{2015}}}\\
&
=\frac{(2^{2^{1}}-1)(2^{2^{1}}+1)\cdots (2^{2^{2015}}+1)}{(2^{2^{0}}-1)2^{2^{0}+\cdots +2^{2015}}}\\
&
=\frac{(2^{2^{2}}-1)(2^{2^{2}}+1)\cdots (2^{2^{2015}}+1)}{2^{2^{0}+\cdots +2^{2015}}}\\
&
\cdots\\
&
=\frac{2^{2^{2016}}-1}{2^{2^{2016}-1}}
\end{align*}
</s>