A sum of positive (usually) distinct unit fractions. The famous Rhind papyrus, dated to around 1650 BC contains a table of representations of as Egyptian fractions for odd n between 5 and 101. The reason the Egyptians chose this method for representing fractions is not clear, although André Weil characterized the decision as "a wrong turn" (Hoffman 1998, pp. 153-154). The unique fraction that the Egyptians did not represent using unit fractions was 2/3 (Wells 1986, p. 29).

Egyptian fractions are almost always required to exclude repeated terms, since representations such as are trivial. Any rational number has representations as an Egyptian fraction with arbitrarily many terms and with arbitrarily large denominators, although for a given fixed number of terms, there are only finitely many. Fibonacci proved that any fraction can be represented as a sum of distinct unit fractions (Hoffman 1998, p. 154). An infinite chain of unit fractions can be constructed using the identity

Martin (1999) showed that for every positive rational number, there exist Egyptian fractions whose largest denominator is at most N and whose denominators form a positive proportion of the integers up to N for sufficiently large N. Each fraction with y odd has an Egyptian fraction in which each denominator is odd (Breusch 1954; Guy 1994, p. 160). Every has a t-term representation where (Vose 1985).

No algorithm is known for producing unit fraction representations having either a minimum number of terms or smallest possible denominator (Hoffman 1998, p. 155). However, there are a number of algorithms (including the binary remainder method, continued fraction unit fraction algorithm, generalized remainder method, greedy algorithm, reverse greedy algorithm, small multiple method, and splitting algorithm) for decomposing an arbitrary fraction into unit fractions. In 1202, Fibonacci published an algorithm for constructing unit fraction representations, and this algorithm was subsequently rediscovered by Sylvester (Hoffman 1998, p. 154; Martin 1999).

Taking the fractions 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, ... (the numerators of which are Sloane's A002260, and the denominators of which are copies of the integer n), the unit fraction representations using the greedy algorithm are

The number of terms in these representations are 1, 1, 2, 1, 1, 2, 1, 2, 2, 3, 1, ... (Sloane's A050205). The minimum denominators for each representation are given by 2, 3, 2, 4, 2, 2, 5, 3, 2, 2, 6, 3, 2, ... (Sloane's A050206), and the maximum denominators are 2, 3, 6, 4, 2, 4, 5, 15, 10, 20, 6, 3, 2, ... (Sloane's A050210).

Wilf posed as a problem that any fraction with odd denominator can be represented as a sum of unit fractions, each having an odd denominator, and Graham proved that infinitely many fractions with a certain range can be represented as a sum of units fractions with square denominators (Hoffman 1998, p. 156).

Paul Erdos and E. G. Straus have conjectured that the Diophantine equation

always can be solved, an assertion sometimes known as the Erdos-Straus conjecture, and Sierpinski (1956) conjectured that

can be solved (Guy 1994).

The harmonic number is never an integer except for . This result was proved in 1915 by Taeisinger, and the more general results that any number of consecutive terms not necessarily starting with 1 never sum to an integer was proved by Kürschák in 1918 (Hoffman 1998, p. 157). In 1932, Erdos proved that the sum of the reciprocals of any number of equally spaced integers is never a reciprocal.

 

Akhmim Wooden Tablet, Egyptian Mathematical Leather Roll, Egyptian Number, Erdos-Straus Conjecture, Harmonic Number, Rhind Papyrus, Unit Fraction