Kummer's Theorem

Theorem

Let $p$ be a prime number.

Let $a, b \in \Z_{\ge 0}$.

Let:

$p^n \divides \dbinom {a + b} b$

but

$p^{n + 1} \nmid \dbinom {a + b} b$

where:

$\divides$ denotes divisibility

$\nmid$ denotes non-divisibility

$\dbinom {a + b} b$ denotes a binomial coefficient.

Then $n$ equals the number of carries that occur when $a$ is added to $b$ using the classical addition algorithm in base $p$.

Proof

This theorem requires a proof. In particular: We need to establish the classical algorithms and build up the knowledge of exactly what a carry is before tackling this delightful little number. 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.

Source of Name

This entry was named for Ernst Eduard Kummer.

Sources

• 1852: Ernst Eduard Kummer: Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen (J. reine angew. Math. Vol. 44: pp. 93 – 146)

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $11$