# Fermat number

A **Fermat number** is a number of the form where is a nonnegative integer.

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

### Proof

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.

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