Let [itex]k\in\mathbb{N}[/itex], and [itex]q[/itex] be a prime factor of [itex]F_{k}=2^{2^{k}}+1[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

Deduce that gcd[itex](q-1,2^{k+1})=2^{k+1}[/itex].

[itex]q|F_{k}[/itex] [itex] \Rightarrow [/itex] [itex] mq = 2^{2^{k}}+1[/itex] for some [itex]m\in\mathbb{N}[/itex]

[itex] 2^{2^{k}}=q-1+(m-1)q[/itex] [itex] \Rightarrow [/itex] [itex] 2^{2^{k}}=q-1[/itex] (mod [itex]q[/itex])

[itex]2^{k+1}|2^{2^{k}}[/itex] since [itex]k+1\leq 2^{k}, \forall k\in \mathbb{N}[/itex]

So [itex]2^{2^{k}}=n2^{k+1}[/itex] for some [itex]n\in \mathbb{N}[/itex].

I think I'm missing something, so any nudge in the right direction would be much appreciated.

Thanks

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Fermat Numbers - Factor Form Proof

**Physics Forums | Science Articles, Homework Help, Discussion**