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method for representing rational
numbers as sums of unit fractions using
practical numbers∗
PrimeFan†
2013-03-22 1:08:08
Fibonacci’s application for practical numbers n was an algorithm to represent
X di
, with the di
(with
m
>
1)
as
sums
of
unit
fractions
proper fractions m
n
n
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
m X di
=
.
n
n
To illustrate the algorithm, let’s rewrite 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
2
14 21
+
+ ,
42 42 42
which we then reduce to lowest terms:
1
1 1
+ + ,
21 3 2
giving us the desired unit fractions.
∗ hMethodForRepresentingRationalNumbersAsSumsOfUnitFractionsUsingPracticalNumbersi
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h11A25i
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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
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