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

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