Egyptian FractionAn 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 Egyptian fractions are almost always required to exclude repeated terms, since representations such as
Martin (1999) showed that for every positive rational number, there exist Egyptian fractions whose largest denominator is at most 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
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). The Egyptian fractions for various constants using the greedy algorithm are summarized in the following table.
Any fraction with odd denominator can be represented as a finite sum of unit fractions, each having an odd denominator (Starke 1952, Breusch 1954). 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 Erdős and E. G. Straus have conjectured that the Diophantine equation
always can be solved, an assertion sometimes known as the Erdős-Straus conjecture, and Sierpiński (1956) conjectured that
can be solved (Guy 1994). The harmonic number |
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).
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
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).
copies of the integer 
), the unit fraction representations using the greedy algorithm are
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, Erdős proved that the sum of the reciprocals of any number of equally spaced integers is never a reciprocal.