# Show that Fermat number $F_n$ and its index $n$ are coprime.

I want to show that $\gcd(F_n,n)=1$, where $F_n=2^{2^n}+1$. How to prove this?

I can show that that $\gcd(F_n, F_m)=1$ for any natural $n$ and $m$, and that $F_{n+1}=(F_n)^2-2F_n+2=F_0\dots F_{n-1}+2$, but I can't see how I can apply this to my problem. What am I missing?

• Why do you think the facts you mention can be applied to this problem? – Chris Eagle Feb 16 '13 at 11:04
• I din't really say that. These are just some basic things I figured out so I thought there might be some use to them. But I haven't found a way to solve this problem either with these tools or in some other way unfortunately. – Ksenia Feb 16 '13 at 11:08

## 1 Answer

Suppose $p$ is a prime dividing $2^{2^n}+1$. Then $2^{2^n}\cong -1 \pmod{p}$, and so $2$ has multiplicative order $2^{n+1}$ mod $p$. Thus $2^{n+1}\mid p-1$, and in particular $p$ is much bigger than $n$.

• +1. Can't resist pointing out that for $n\ge2$ your argument shows first that $p\equiv1\pmod8$. Thus $2$ is a quadratic residue modulo $p$, and we can further conclude that $2^{n+2}\mid p-1$. – Jyrki Lahtonen Feb 16 '13 at 11:21
• We haven't studied multiplicative order yet and I can't see where $2^{n+1}\mod p-1$ comes from... WHy is it so? (sorry it took me so long to answer) – Ksenia Feb 20 '13 at 16:02