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

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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 $p$.


Note that for any integer $x$:

\(\ds x^p - x\) \(\equiv\) \(\ds 0\) \(\ds \pmod p\) Corollary $1$ to Fermat's Little Theorem
\(\ds \) \(\equiv\) \(\ds x^{\overline p}\) \(\ds \pmod p\) Divisibility of Product of Consecutive Integers
\(\ds \) \(\equiv\) \(\ds \sum_k {p \brack k} x^k\) \(\ds \pmod p\) Sum over k of Unsigned Stirling Numbers of First Kind by x^k

By comparing coefficients:

$\ds {p \brack p} \equiv 1 \pmod p$
$\ds {p \brack 1} \equiv -1 \pmod p$
$\ds {p \brack k} \equiv 0 \pmod p$ for $k \ne 1, p$

or in a more compact form:

$\ds {p \brack k} \equiv \delta_{k p} - \delta _{k 1} \pmod p$


$\ds {p \brack k}$ denotes an unsigned Stirling number of the first kind
$\delta$ is the Kronecker delta.

From Harmonic Number as Unsigned Stirling Number of First Kind over Factorial:

$\ds H_{p - 1} = \frac {p \brack 2} {\paren {p - 1}!}$

From the above we have:

$\ds p \divides {p \brack 2}$

By Prime iff Coprime to all Smaller Positive Integers we also have:

$p \nmid \paren {p - 1}!$

Hence the numerator of $H_{p - 1}$, when expressed in canonical form, is divisible by $p$.