FermatNumbersAreCoprime 2004-11-25 19:33:34 2007-08-04 02:25:36 Theorem Fermat numbers \usepackage{amsthm} \newtheorem*{thm*}{Theorem} \begin{thm*} Any two Fermat numbers are coprime. \end{thm*} Proof.\\ Let $F_m$ and $F_n$ two Fermat numbers, and assume $m<n$. Let $d$ a positive common divisor of $F_n$ and $F_m$, that is \[ d \mid F_m,\qquad d\mid F_n. \] If $d\mid F_m$ then $d\mid F_1F_2\cdots F_{n-1}$ since some factor must be $F_m$ itself. But $F_n-F_1F_2\cdots F_{n-1}=2$ 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.