I am confused of a question that needs to know the greatest common divisor of $2^m+1$ and $2^n+1$ ($m,n$ are positive integers), but I don't really know. I am pretty sure that the greatest common divisor of $2^m-1$ and $2^n-1$ ($m,n$ are positive integers) is $2^{\gcd\left(m,n\right)}-1$, even I can prove it by the Euclidean algorithm. However, it is hard to use it in this problem, so I want you guys to help me. Thanks!
P.S.
I created an excel and I observed the answer (maybe?) from it, but I can't prove or disprove it. Here is my conclusion from the excel: $$\gcd\left(2^m+1,2^n+1\right)=\begin{cases} 2^{\gcd\left(m,n\right)}+1 \\ 1 \end{cases}\begin{matrix} \text{when }m,n\text{ contain the exact same power of }2 \\ \text{otherwise} \end{matrix}$$ Hope it will help me and you guys solving this quesion :D
The link of The excel
from Hot Weekly Questions - Mathematics Stack Exchange
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