Egyptian Fraction - Modern Number Theory

Modern Number Theory

For more information on this subject, see Erdős–Graham conjecture, Znám's problem, and Engel expansion.

Modern number theorists have studied many different problems related to Egyptian fractions, including problems of bounding the length or maximum denominator in Egyptian fraction representations, finding expansions of certain special forms or in which the denominators are all of some special type, the termination of various methods for Egyptian fraction expansion, and showing that expansions exist for any sufficiently dense set of sufficiently smooth numbers.

• The Erdős–Graham conjecture in combinatorial number theory states that, if the unit fractions are partitioned into finitely many subsets, then one of the subsets can be used to form an Egyptian fraction representation of one. That is, for every r > 0, and every r-coloring of the integers greater than one, there is a finite monochromatic subset S of these integers such that

The conjecture was proven in 2003 by Ernest S. Croot, III.

• Znám's problem and primary pseudoperfect numbers are closely related to the existence of Egyptian fractions of the form

For instance, the primary pseudoperfect number 1806 is the product of the prime numbers 2, 3, 7, and 43, and gives rise to the Egyptian fraction 1 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1806.

• Egyptian fractions are normally defined as requiring all denominators to be distinct, but this requirement can be relaxed to allow repeated denominators. However, this relaxed form of Egyptian fractions does not allow for any number to be represented using fewer fractions, as any expansion with repeated fractions can be converted to an Egyptian fraction of equal or smaller length by repeated application of the replacement

if k is odd, or simply by replacing 1/k+1/k by 2/k if k is even. This result was first proven by Takenouchi (1921).

• Graham and Jewett (see Wagon 1991 and Beeckmans 1993) proved that it is similarly possible to convert expansions with repeated denominators to (longer) Egyptian fractions, via the replacement

This method can lead to long expansions with large denominators, such as

Botts (1967) had originally used this replacement technique to show that any rational number has Egyptian fraction representations with arbitrarily large minimum denominators.

• Any fraction x/y has an Egyptian fraction representation in which the maximum denominator is bounded by

(Tenenbaum & Yokota 1990) and a representation with at most

terms (Vose 1985).

• Graham (1964) characterized the numbers that can be represented by Egyptian fractions in which all denominators are nth powers. In particular, a rational number q can be represented as an Egyptian fraction with square denominators if and only if q lies in one of the two half-open intervals

• Martin (1999) showed that any rational number has very dense expansions, using a constant fraction of the denominators up to N for any sufficiently large N.
• Engel expansion, sometimes called an Egyptian product, is a form of Egyptian fraction expansion in which each denominator is a multiple of the previous one:

In addition, the sequence of multipliers a_i is required to be nondecreasing. Every rational number has a finite Engel expansion, while irrational numbers have an infinite Engel expansion.

• Anshel & Goldfeld (1991) study numbers that have multiple distinct Egyptian fraction representations with the same number of terms and the same product of denominators; for instance, one of the examples they supply is

Unlike the ancient Egyptians, they allow denominators to be repeated in these expansions. They apply their results for this problem to the characterization of free products of Abelian groups by a small number of numerical parameters: the rank of the commutator subgroup, the number of terms in the free product, and the product of the orders of the factors.