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 ${\displaystyle \frac{m}{n}}={\displaystyle \sum \frac{d_i}{n}}$.

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.