Representation of 1 as Sum of n Unit Fractions
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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:
$n$ $U \left({n}\right)$ $1$ $1$ $2$ $1$ $3$ $3$ $4$ $14$ $5$ $147$ $6$ $3462$
This sequence is A002966 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
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: $U \left({2}\right) = 1$
From Sum of 3 Unit Fractions that equals 1:
- $U \left({3}\right) = 3$
From Sum of 4 Unit Fractions that equals 1:
- $U \left({4}\right) = 14$
From Sum of 5 Unit Fractions that equals 1:
- $U \left({5}\right) = 147$
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Historical Note
According to 1997: David Wells: Curious and Interesting Numbers (2nd ed.), this result is attributed to David Breyer Singmaster.
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $147$