# 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. 1.

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

2. 2.

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

3. 3.

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

4. 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
Title method for representing rational numbers as sums of unit fractions using practical numbers MethodForRepresentingRationalNumbersAsSumsOfUnitFractionsUsingPracticalNumbers 2013-03-22 18:07:00 2013-03-22 18:07:00 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Algorithm msc 11A25