# Definition:Fermat Number/Naming Conventions

## Definition

The **Fermat number** $F_0$ is often referred to as the ** $1$st Fermat number**, so (confusingly) this convention dictates that $F_n$ is the **$n + 1$th Fermat number**.

However, another convention is that $F_0$ can be referred to as the **zeroth Fermat number**, thus bringing the appellation in line such that $F_n$ is the **$n$th Fermat number**.

Both conventions are in place, sometimes in the same work.

For example, David Wells, in his *Curious and Interesting Numbers, 2nd ed.* of $1997$, refers to $5 = F_1$ in Section $5$ as the **$2$nd Fermat number**.

However, in Section $257$ he defines $F_3 = 2^{2^3} + 1 = 257$ as the **$3$rd Fermat number**.

Similarly, in Section $65,537$ he defines $F_4 = 2^{2^4} + 1 = 65 \, 537$ as the **$4$th Fermat number**, and so on.

Both of these naming conventions is more or less clumsy.

$\mathsf{Pr} \infty \mathsf{fWiki}$ takes the position that the cat has to jump one way or the other, and so uses the second of these conventions:

- $F_n$ is the
**$n$th Fermat number**.