 Egyptian fraction expansion

An Egyptian fraction is 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 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

(1)
Martin (1999) showed that for every positive rational number, there exist Egyptian fractions whose largest denominator is at most and whose denominators form a positive proportion of the integers up to for sufficiently large . Each fraction with odd has an Egyptian fraction in which each denominator is odd (Breusch 1954; Guy 1994, p. 160). Every has a -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). Mathword 