Practical Numbers and Egyptian Fractions
If n is practical, then any rational number of the form m/n may be represented as a sum ∑di/n where each di is a distinct divisor of n. Each term in this sum simplifies to a unit fraction, so such a sum provides a representation of m/n as an Egyptian fraction. For instance,
Fibonacci, in his 1202 book Liber Abaci lists several methods for finding Egyptian fraction representations of a rational number. Of these, the first is to test whether the number is itself already a unit fraction, but the second is to search for a representation of the numerator as a sum of divisors of the denominator, as described above; this method is only guaranteed to succeed for denominators that are practical. Fibonacci provides tables of these representations for fractions having as denominators the practical numbers 6, 8, 12, 20, 24, 60, and 100.
Vose (1985) showed that every number x/y has an Egyptian fraction representation with terms. The proof involves finding a sequence of practical numbers ni with the property that every number less than ni may be written as a sum of distinct divisors of ni. Then, i is chosen so that ni − 1 < y ≤ ni, and xni is divided by y giving quotient q and remainder r. It follows from these choices that . Expanding both numerators on the right hand side of this formula into sums of divisors of ni results in the desired Egyptian fraction representation. Tenenbaum & Yokota (1990) use a similar technique involving a different sequence of practical numbers to show that every number x/y has an Egyptian fraction representation in which the largest denominator is .
Read more about this topic: Practical Number
Famous quotes containing the words practical, numbers and/or egyptian:
“Juggling produces both practical and psychological benefits.... A woman’s involvement in one role can enhance her functioning in another. Being a wife can make it easier to work outside the home. Being a mother can facilitate the activities and foster the skills of the efficient wife or of the effective worker. And employment outside the home can contribute in substantial, practical ways to how one works within the home, as a spouse and as a parent.”
—Faye J. Crosby (20th century)
“The barriers of conventionality have been raised so high, and so strangely cemented by long existence, that the only hope of overthrowing them exists in the union of numbers linked together by common opinion and effort ... the united watchword of thousands would strike at the foundation of the false system and annihilate it.”
—Mme. Ellen Louise Demorest 1824–1898, U.S. women’s magazine editor and woman’s club movement pioneer. Demorest’s Illustrated Monthly and Mirror of Fashions, p. 203 (January 1870)
“What was I saying? An Egyptian king
Once touched long fingers, which are not anything.”
—Allen Tate (1899–1979)