Numerator of p-1th Harmonic Number is Divisible by p^2 for Prime Greater than 3

Theorem

Let $p$ be a prime number such that $p > 3$.

Consider the harmonic number $H_{p - 1}$ expressed in canonical form.

The numerator of $H_{p - 1}$ is divisible by $p^2$.

Proof

This theorem requires a proof. In particular: research and invoke Wolstenholme's Theorem You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1979: G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers (5th ed.): $7.8$: A theorem of Wolstenholme: Theorem $116$
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: Exercise $17$ (Solution)