# 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 $p$.

## Source of Name

This entry was named for Ernst Eduard Kummer.