Let $m, n$ be positive integers, with $m > n$ and $2^{2^m}+1$ and $2^{2^n}+1$, Fermat numbers.
To prove that both Fermat numbers are coprime, it's sufficient to state $$\begin{align}(2^{2^m}+1, 2^{2^n}+1) =&\\ (2^{2^m}+1-(2^{2^n}+1), 2^{2^n}+1) =& \\(2^{2^m}-2^{2^n}, 2^{2^n}+1) =&\\ (2k,2q+1) \overset{(1)}{=}& \quad 1\end{align}$$.
Where $2k = 2^{2^m}-2^{2^n}$ and $2q+1=2^{2^n}+1$.
$(a,b)$ is a shorthand notation for $gcd(a,b)$.
Is this proof incorrect? Why so?
In particular, is the setp $(1)$ wrong?
I'm unable to see what I'm missing here.
Thanks in advance.
Thanks to @GeorgeIvey. Certainly assuming that for every $q,k$ we have $(2q,2k+1)=1$ is false, since $(42,3) = 3$, since $42 = 2 \cdot 7 \cdot 3$.