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

## 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 1

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 $k'= - \dfrac {\paren {p - 1}!} k \bmod p$.

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

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 | |||||||||

\(\displaystyle \) | \(\equiv\) | \(\displaystyle 0\) | \(\displaystyle \pmod p\) |

$\blacksquare$

## Proof 2

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

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

A particular theorem is missing. In particular: The above is Knuth 4.6.2-6 |

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.

## Historical Note

Donald E. Knuth reports in his *The Art of Computer Programming: Volume 1: Fundamental Algorithms, 3rd ed.* ($1997$) that this result was established by Edward Waring in $1782$, but he gives no further details.

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: Exercise $17$