Egyptian Fraction

An Egyptian fraction is a sum of positive unit fractions, usually required to be distinct. 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 positive 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 positive fraction can be represented as a sum of distinct unit fractions (Hoffman 1998, p. 154). An infinite chain of unit fractions can be constructed by repeatedly splitting the term with largest denominator using the identity

(1)

for integers .

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 positive fraction with odd has an Egyptian fraction in which each denominator is odd (Starke 1952, Breusch 1954; Guy 1994, p. 160). For integers , there is a -term representation with as (Vose 1985).

A minimum number of terms can be found by finite exhaustive search. For a positive remainder represented by distinct unit fractions in increasing order of denominators, the first denominator satisfies , where is the ceiling function and is the floor function. Recursion therefore gives a finite search for each fixed ; testing successive eventually finds a shortest representation. The smallest possible largest denominator can likewise be found by enumerating subsets of for successive (Eppstein). There are also 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 positive fraction into unit fractions. According to Martin (1999, p. 3641), Fibonacci published an algorithm for constructing unit fraction representations in 1202, and this algorithm was subsequently rediscovered by Sylvester.

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

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)

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

The Egyptian fractions for the fractional parts of various constants using the greedy algorithm are summarized in the following table, where denotes Catalan's constant.

constant	OEIS	Egyptian fraction for
	A006487	3, 13, 253, 218201, 61323543802, ...
	A118325	2, 5, 32, 1249, 5986000, 438522193400489, ...
	A069139	2, 5, 141, 68575, 32089377154, ...
	A006525	2, 5, 55, 9999, 3620211523, 25838201785967533906, ...
	A006526	3, 29, 15786, 513429610, 339840390654894740, ...
	A110820	2, 13, 3418, 52016149, 153922786652714666, ...
	A118323	2, 3, 13, 176, 36543, ...
	A117116	2, 9, 145, 37986, 2345721887, ...
	A118324	2, 6, 38, 6071, 144715221, ...
	A001466	8, 61, 5020, 128541455, 162924332716605980, ...
	A006524	4, 15, 609, 845029, 1010073215739, ...

Graham proved that infinitely many fractions in a certain range can be represented as a sum of unit fractions with square denominators (Hoffman 1998, p. 156).

The Erdős-Straus conjecture asserts that the Diophantine equation

(12)

has a solution in positive integers , , and for every integer , without requiring distinct denominators. Similarly, Sierpiński (1956) conjectured that

(13)

can be solved under the same conditions (Guy 1994).

The harmonic number is never an integer except for . According to Hoffman (1998, p. 157), Taeisinger proved this result in 1915, and Kürschák proved in 1918 the more general result that a sum of at least two reciprocals of consecutive positive integers never equals an integer. Equally spaced denominators can give a unit fraction; for example, .

Nontrivial sets of integers are known whose reciprocals sum to small integers. For example, there exists a set of 366 positive integers (with maximum 992) whose sum of reciprocals is exactly 2 (Mackenzie 1997; Martin). A similar set of 453 small positive integers is known whose reciprocals sum to 6 (Martin).