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

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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

From Polynomial x^p - x is Congruent mod p to x to the p-1 Rising:

$x^{\overline p} \equiv x^p - x$

Thus from Sum over k of Unsigned Stirling Numbers of First Kind by x^k:

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

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.



Sources