Fermat numbers are coprime

Theorem.

Any two Fermat numbers are coprime.

Proof.
Let $F_m$ and $F_n$ two Fermat numbers, and assume .Let $d$ a positive common divisor of $F_n$ and $F_m$, that is

$$d\mid F_m,d\mid F_n.$$

If $d\mid F_m$ then $d\mid F_1{F}_2\mathrm{\cdots }{F}_{n-1}$since some factor must be $F_m$ itself.But $F_n-F_1{F}_2\mathrm{\cdots }{F}_{n-1}=2$ and so $d\mid 2$.Since $d$ is odd, we must have $d=1$.

Therefore, the greatest common divisor of any two Fermat numbers must be $1$.

Q.E.D.