16 questions linked to/from Fermat numbers are coprime
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### Prove that if $a,m,n \in \mathbb{Z}^+$ and $m \ne n$ then $\gcd(a^{2^m}+1,a^{2^n}+1)$ is 1 if a is even and 2 if a is odd. [duplicate]

I know that if a pime q|a^2^m+1 and q divides a^2^n+1 then q divides their sum and difference but i don't know how to proceed further. please help
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### Problems regarding GCD [duplicate]

If $m > n$ and $a,m,n$ are positive, with $m$ not equal to $n$, find the greatest common divisor of $2^{2^m}+1, 2^{2^n}+1$. Please solve this problem using Euclid's algorithm. I tried to use ...
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### Show that the only divisors are $1$ and $2$ for $(z^{(2^x)}+1)$ and $(z^{(2^y)}+1)$ where $x,y,z\in\mathbb{N}$

I am trying to show that the only divisors are $1$ and $2$ for both $(z^{(2^x)}+1)$ and $(z^{(2^y)}+1)$ where $x,y,z\in\mathbb{N}$. To start the problem, the logical choice is to use difference of ...
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### If $m\neq n$ what is $\mathrm{gcd}(a^{2n}+1,a^{2m}+1)?$ [closed]

If $m\neq n,$ compute $\mathrm{gcd}(a^{2n}+1,a^{2m}+1).$ In my question, $m$ , $n$ , and $a$ are positive integers.
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### Finding the HCF

Find $(a^{2^m}+1, a^{2^n}+1)$ when a is odd and a,m,n are positive integers and m is not equal to n. I know that the hcf is a multiple of two but I can't prove that it is 2 which is the answer. Plz ...