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.