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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 created: h2013-03-2i by: hPrimeFani version: h40663i Privacy setting: h1i hAlgorithmi h11A25i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 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 2