# Representation of 1 as Sum of n Unit Fractions

## Theorem

Let $U \left({n}\right)$ denote the number of different ways of representing $1$ as the sum of $n$ unit fractions.

Then for various $n$, $U \left({n}\right)$ is given by the following table:

$n$ $U \left({n}\right)$
$1$ $1$
$2$ $1$
$3$ $3$
$4$ $14$
$5$ $147$
$6$ $3462$

## Proof

Trivially:

$1 = \dfrac 1 1$

and it follows that: $U \left({1}\right) = 1$

Also trivially:

$1 = \dfrac 1 2 + \dfrac 1 2$

and it follows that: $U \left({2}\right) = 1$

$U \left({3}\right) = 3$
$U \left({4}\right) = 14$
$U \left({5}\right) = 147$

## Historical Note

According to 1997: David Wells: Curious and Interesting Numbers (2nd ed.), this result is attributed to David Breyer Singmaster.