The first five Fermat numbers (for ) are A prime Fermat number is known as a Fermat prime. Each of the first five Fermat numbers is a Fermat prime. Based on these results, one might conjecture (as did Fermat) that all Fermat numbers are prime. However, this fails for : . In fact, the primes listed above are the only Fermat numbers known to be prime. The Fermat number is known to be composite for .
Relative primality of Fermat numbers
The Fermat numbers and are relatively prime for all
We prove that the Fermat numbers satisfy the following recursion, from which the claimed result will follow: for all ,
We proceed by induction. For we have , as desired. Suppose that . Now observe that as desired.
It follows that if that , so and are relatively prime, as claimed.
This article is a stub. Help us out by.