Practical Number - Practical Numbers and Egyptian Fractions

Practical Numbers and Egyptian Fractions

If n is practical, then any rational number of the form m/n may be represented as a sum ∑d_i/n where each d_i 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 n_i with the property that every number less than n_i may be written as a sum of distinct divisors of n_i. Then, i is chosen so that n_{i − 1} < y ≤ n_i, and xn_i 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 n_i 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 .