Numerator of p-1th Harmonic Number is Divisible by Prime p/Proof 2

Theorem

Let $p$ be an odd prime.

Consider the harmonic number $H_{p - 1}$ expressed in canonical form.

The numerator of $H_{p - 1}$ is divisible by $p$.

Proof

$x^{\overline p} \equiv x^p - x$
$\displaystyle \left[{p \atop k}\right] \equiv \delta_{k p} - \delta _{k 1}$

where:

$\displaystyle \left[{p \atop k}\right]$ denotes an unsigned Stirling number of the first kind
$\delta$ is the Kronecker delta.

The result follows from Harmonic Number as Unsigned Stirling Number of First Kind over Factorial.