Fermat numbers are coprime

Theorem.

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 $d\mid F_{1}F_{2}\cdots F_{n-1}$ since some factor must be $F_{m}$ itself. But $F_{n}-F_{1}F_{2}\cdots F_{n-1}=2$ 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.

Title	Fermat numbers are coprime
Canonical name	FermatNumbersAreCoprime
Date of creation	2013-03-22 14:51:24
Last modified on	2013-03-22 14:51:24
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	5
Author	yark (2760)
Entry type	Theorem
Classification	msc 11A51