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

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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

Add the terms of $H_{p - 1}$ using the definition of rational addition to obtain $\dfrac m n$.

Do not cancel common prime factors from $m$ and $n$.

It is seen that $n = \paren {p - 1}!$

Hence $p$ is not a divisor of $n$.


The numerator $m$ is seen to be:

$m = \dfrac {\paren {p - 1}!} 1 + \dfrac {\paren {p - 1}!} 2 + \cdots + \dfrac {\paren {p - 1}!} {p - 1}$

Thus it is sufficient to show that $m$ is a multiple of $p$.

Each term in this sum is an integer of the form $\dfrac {\paren {p - 1}!} k$.

For each $k \in \set {1, 2, \ldots, p - 1}$, define $k'= - \dfrac {\paren {p - 1}!} k \bmod p$.

By Wilson's Theorem

$k k' \equiv -\paren {p - 1}! \equiv 1 \pmod p$

Therefore

$k' \equiv k^{-1} \pmod p$

From the corollary to Reduced Residue System under Multiplication forms Abelian Group:

$\struct {\Z'_p, \times}$ is an abelian group.

Since Inverse in Group is Unique, the set:

$\set {1', 2', \ldots, \paren {p - 1}'}$

is merely the set:

$\set {1, 2, \ldots, p - 1}$

in a different order.

Thus

\(\displaystyle m\) \(=\) \(\displaystyle \dfrac {\paren {p - 1}!} 1 + \dfrac {\paren {p - 1}!} 2 + \cdots + \dfrac {\paren {p - 1}!} {p - 1}\)
\(\displaystyle \) \(\equiv\) \(\displaystyle 1 + 2 + \cdots + p - 1\) \(\displaystyle \pmod p\)
\(\displaystyle \) \(\equiv\) \(\displaystyle \frac {p \paren {p - 1} } 2\) \(\displaystyle \pmod p\) Closed Form for Triangular Numbers
\(\displaystyle \) \(\equiv\) \(\displaystyle 0\) \(\displaystyle \pmod p\)

$\blacksquare$


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