# 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:

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

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.

## 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 $17$