method for representing rational numbers as sums of unit fractions using practical numbers
Fibonacci’s application for practical numbers was an algorithm to represent proper fractions (with ) as sums of unit fractions , with the being divisors of the practical number . (By the way, there are infinitely many practical numbers which are also Fibonacci numbers). The method is:
Reduce the fraction to lowest terms. If the numerator is then 1, we’re done.
Rewrite as a sum of divisors of .
Make those divisors of that add up to into the numerators of fractions with as denominator.
Reduce those fractions to lowest terms, thus obtaining the representation .
To illustrate the algorithm, let’s rewrite 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
which we then reduce to lowest terms:
giving us the desired unit fractions.
- 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|
|Date of creation||2013-03-22 18:07:00|
|Last modified on||2013-03-22 18:07:00|
|Last modified by||PrimeFan (13766)|