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,, in his 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.

takes the position that the cat has to jump one way or the other, and so uses the first of these conventions:
 * $F_n$ is the $n + 1$th Fermat number.