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Homemethod for representing rational numbers as sums of unit fractions using practical numbers

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# method for representing rational numbers as sums of unit fractions using practical numbers

Fibonacci’s application for practical numbers $n$ was an algorithm to represent proper fractions $\frac{m}{n}$ (with $m>1$) as sums of unit fractions $\displaystyle\sum\frac{d_{i}}{n}$, with the $d_{i}$ being divisors of the practical number $n$. (By the way, there are infinitely many practical numbers which are also Fibonacci numbers). The method is:

1. Reduce the fraction to lowest terms. If the numerator is then 1, we’re done.

2. Rewrite $m$ as a sum of divisors of $n$.

3. Make those divisors of $n$ that add up to $m$ into the numerators of fractions with $n$ as denominator.

4. Reduce those fractions to lowest terms, thus obtaining the representation $\displaystyle\frac{m}{n}=\sum\frac{d_{i}}{n}$.

To illustrate the algorithm, let’s rewrite $\frac{37}{42}$ as a sum of unit fractions. Since 42 is practical, success is guaranteed.

At the first step we can’t reduce this fraction because 37 is a prime number. So we go on to the second step, and represent 37 as 2 + 14 + 21. This gives us the fractions

$\frac{2}{42}+\frac{14}{42}+\frac{21}{42},$ |

which we then reduce to lowest terms:

$\frac{1}{21}+\frac{1}{3}+\frac{1}{2},$ |

giving us the desired unit fractions.

# References

- 1 M. R. Heyworth, “More on panarithmic numbers” New Zealand Math. Mag. 17 (1980): 28 - 34
- 2 Giuseppe Melfi, “A survey on practical numbers” Rend. Sem. Mat. Univ. Pol. Torino 53 (1995): 347 - 359

## Mathematics Subject Classification

11A25*no label found*

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