Egyptian fraction

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An Egyptian fraction is the sum of distinct unit fractions, such as $\tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16}$ . That is, each fraction in the series has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The sum of an expression of this type is a positive rational number a/b; for instance the Egyptian fraction above sums to 43/48. It can be shown that every positive rational number can be represented in this way by an Egyptian fraction. This type of sum, and similar sums also including 2/3 as a summand, were used as a serious notation for fractions by the ancient Egyptians, continuing into medieval times. In modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern number theory and recreational mathematics, as well as in modern historical studies of ancient mathematics.

1 Ancient Egypt
2 Medieval mathematics
3 Modern number theory
4 Open problems
5 Bibliography
6 External links

[edit] Ancient Egypt

For more information on this subject, see Egyptian numerals, Eye of Horus, and Egyptian mathematics.

Egyptian fraction notation was developed in the Middle Kingdom of Egypt, altering the Old Kingdom's Eye of Horus numeration system. Five early texts in which Egyptian fractions appear were the Egyptian Mathematical Leather Roll, the Moscow Mathematical Papyrus, the Reisner Papyrus, the Kahun Papyrus and the Akhmim Wooden Tablet. A later text, the Rhind Mathematical Papyrus, introduced improved ways of writing Egyptian fractions. The Rhind papyrus was written by Ahmes and dates from the Second Intermediate Period; it includes a table of Egyptian fraction expansions for rational numbers 2/n, as well as 84 word problems. Solutions to each problem were written out in scribal shorthand, with the final answers of all 84 problems being expressed in Egyptian fraction notation. 2/n tables similar to the one on the Rhind papyrus also appear on some of the other texts.

To write the unit fractions used in their Egyptian fraction notation, in hieroglyph script, the Egyptians placed the hieroglyph

(er, "[one] among" or possibly re, mouth) above a number to represent the reciprocal of that number. Similarly in hieratic script they drew a line over the letter representing the number. For example:

$= \frac{1}{3}$

$= \frac{1}{10}$

The Egyptians had special symbols for 1/2, 2/3, and 3/4 that were used to reduce the size of numbers greater than 1/2 when such numbers were converted to an Egyptian fraction series. The remaining number after subtracting one of these special fractions was written using the usual Egyptian fraction representations.

$= \frac{1}{2}$

$= \frac{2}{3}$

$= \frac{3}{4}$

The Egyptians also used an alternative notation modified from the Old Kingdom and based on the parts of the Eye of Horus to denote a special set of fractions of the form 1/2^k (for k = 1, 2, ..., 6), that is, dyadic rational numbers. These "Horus-Eye fractions" were used in the Middle Kingdom in conjunction with the later notation for Egyptian fractions to subdivide a hekat, the primary ancient Egyptian volume measure for grain, bread, and other small quantities of volume, the subject of the Akhmim Wooden Tablet. If any remainder was left after expressing a quantity in Eye of Horus fractions of a hekat, the remainder was written using the regular Egyptian fraction notation as multiples of a ro, a unit equal to 1/320 of a hekat.

Modern historians of mathematics have studied the Rhind papyrus and other ancient sources in an attempt to discover the methods the Egyptians used in calculating with Egyptian fractions. In particular, study in this area has concentrated on understanding the tables of expansions for numbers of the form 2/n in the Rhind papyrus. Although these expansions can generally be described as algebraic identities, they do not match any single identity; rather, different methods were used for prime and for composite denominators, and more than one method was used for numbers of each type:

For small odd prime denominators p, the expansion 2/p = 2/(p + 1) + 2/p(p + 1) was used.

For larger prime denominators, an expansion of the form 2/p = 1/A + (2A-p)/Ap was used, where A is a number with many divisors (such as a practical number) in the range p/2 < A < p. The remaining term (2A-p)/Ap was expanded by representing the number 2A-p as a sum of divisors of A and forming a fraction d/Ap for each such divisor d in this sum (Hultsch 1895, Bruins 1957). As an example, Ahmes' expansion 2/37 = 1/24 + 1/111 + 1/296 fits this pattern, with A = 24 and 2A-p = 11 = 3+8, since 3 and 8 are divisors of 24. There may be many different expansions of this type for a given p; however, as K. S. Brown observed, the expansion chosen by the Egyptians was often the one that caused the largest denominator to be as small as possible, among all expansions fitting this pattern.

For composite denominators, factored as p×q, one can expand 2/pq using the identity 2/pq = 1/aq + 1/apq, where a = (p+1)/2. For instance, applying this method for pq = 21 gives p = 3, q = 7, and a = (3+1)/2 = 2, producing the expansion 2/21 = 1/14 + 1/42 from the Rhind papyrus. Some authors have preferred to write this expansion as 2/A × A/pq, where A = p+1 (Gardner, 2002); replacing the second term of this product by p/pq + 1/pq, applying the distributive law to the product, and simplifying leads to an expression equivalent to the first expansion described here. This method appears to have been used for many of the composite numbers in the Rhind papyrus (Gillings 1982, Gardner 2002), but there are exceptions, notably 2/35, 2/91, and 2/95 (Knorr 1982).

One can also expand 2/pq as 1/pr + 1/qr, where r = (p+q)/2. For instance, Ahmes expands 2/35 = 1/30 + 1/42, where p = 5, q = 7, and r = (5+7)/2 = 6. Later scribes used a more general form of this expansion, n/pq = 1/pr + 1/qr, where r =(p + q)/n, which works when p ≡ -q (mod n) (Eves, 1953).

For some other composite denominators, the expansion for 2/pq has the form of an expansion for 2/q with each denominator multiplied by p. For instance, 95=5×19, and 2/19 = 1/12 + 1/76 + 1/114 (as can be found using the method for primes with A = 12), so 2/95 = 1/(5×12) + 1/(5×76) + 1/(5×114) = 1/60 + 1/380 + 1/570 (Eves, 1953). This expression can be simplified as 1/380 + 1/570 = 1/228 but the Rhind papyrus uses the unsimplified form.

The final (prime) expansion in the Rhind papyrus, 2/101, does not fit any of these forms, but instead uses an expansion 2/p = 1/p + 1/2p + 1/3p + 1/6p that may be applied regardless of the value of p. That is, 2/101 = 1/101 + 1/202 + 1/303 + 1/606. A related expansion was also used in the Egyptian Mathematical Leather Roll for several cases.

[edit] Medieval mathematics

For more information on this subject, see Liber Abaci and Greedy algorithm for Egyptian fractions.

Egyptian fraction notation continued to be used in Greek times and into the middle ages (Struik 1967), despite complaints as early as Ptolemy's Almagest about the clumsiness of the notation compared to alternatives such as the Babylonian base-60 notation. An important text of medieval mathematics, the Liber Abaci (1202) of Leonardo of Pisa (more commonly known as Fibonacci), gives us some insight into the uses of Egyptian fractions in the middle ages, and introduces topics that continue to be important in modern mathematical study of these series.

The primary subject of the Liber Abaci is calculations involving decimal and vulgar fraction notation, which eventually replaced Egyptian fractions. Fibonacci himself used a complex notation for fractions involving a combination of a mixed radix notation with sums of fractions. Many of the calculations throughout Fibonacci's book involve numbers represented as Egyptian fractions, and one section of this book (Sigler 2002, chapter II.7) provides a list of method for conversion of vulgar fractions to Egyptian fractions. If the number is not already a unit fraction, the first method in this list is to attempt to split the numerator into a sum of divisors of the denominator; this is possible whenever the denominator is a practical number, and Liber Abaci includes tables of expansions of this type for the practical numbers 6, 8, 12, 20, 24, 60, and 100.

The next several methods involve algebraic identities such as $\tfrac{a}{ab-1}=\tfrac{1}{b}+\tfrac{1}{b(ab-1)}.$ For instance, Fibonacci represents the fraction $\tfrac{8}{11}$ by splitting the numerator into a sum of two numbers, each of which divides one plus the denominator: $\tfrac{8}{11}=\tfrac{6}{11}+\tfrac{2}{11}.$ Fibonacci applies the algebraic identity above to each these two parts, producing the expansion $\tfrac{8}{11}=\tfrac{1}{2}+\tfrac{1}{22}+\tfrac{1}{6}+\tfrac{1}{66}.$ Fibonacci describes similar methods for denominators that are two or three less than a number with many factors.

In the rare case that these other methods all fail, Fibonacci suggests a greedy algorithm for computing Egyptian fractions, in which one repeatedly chooses the unit fraction with the smallest denominator that is no larger than the remaining fraction to be expanded: that is, in more modern notation, we replace a fraction x/y by the expansion

$\frac{x}{y}=\frac{1}{\lceil y/x\rceil}+\frac{-y\,\bmod\, x}{y\lceil y/x\rceil}.$

Fibonacci suggests switching to another method after the first such expansion, but he also gives examples in which this greedy expansion was iterated until a complete Egyptian fraction expansion was constructed: $\tfrac{4}{13}=\tfrac{1}{4}+\tfrac{1}{18}+\tfrac{1}{468}$ and $\tfrac{17}{29}=\tfrac{1}{2}+\tfrac{1}{12}+\tfrac{1}{348}.$

As later mathematicians showed, each greedy expansion reduces the numerator of the remaining fraction to be expanded, so this method always terminates with a finite expansion. However, compared to ancient Egyptian expansions or to more modern methods, this method may produce expansions that are quite long, with large denominators, and Fibonacci himself noted the awkwardness of the expansions produced by this method. For instance, the greedy method expands

$\frac{5}{121}=\frac{1}{25}+\frac{1}{757}+\frac{1}{763309}+\frac{1}{873960180913}+\frac{1}{1527612795642093418846225},$

while other methods lead to the much better expansion

$\frac{5}{121}=\frac{1}{33}+\frac{1}{121}+\frac{1}{363}.$

Sylvester's sequence 2, 3, 7, 43, 1807, ... can be viewed as generated by an infinite greedy expansion of this type for the number one, where at each step we choose the denominator $\lfloor y/x\rfloor+1$ instead of $\lceil y/x\rceil$ , and sometimes Fibonacci's greedy algorithm is attributed to Sylvester.

After his description of the greedy algorithm, Fibonacci suggests yet another method, expanding a fraction $a / b$ by searching for a number c having many divisors, with $b / 2 < c < b$ , replacing $a / b$ by $a c / b c$ , and expanding $a c$ as a sum of divisors of $b c$ , similar to the method proposed by Hultsch and Bruins to explain the ancient RMP 2/p expansions.

[edit] 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

$\sum_{n\in S}1/n = 1.$

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

$\sum\frac1{x_i} + \prod\frac1{x_i}=1.$

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

$\frac1k+\frac1k=\frac2{k+1}+\frac2{k(k+1)}$

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

$\frac1k+\frac1k=\frac1k+\frac1{k+1}+\frac1{k(k+1)}.$

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

$\frac45=\frac15+\frac16+\frac17+\frac18+\frac1{30}+\frac1{31}+\frac1{32}+\frac1{42}+\frac1{43}+\frac1{56}+ \frac1{930}+\frac1{931}+\frac1{992}+\frac1{1806}+\frac1{865830}.$

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

$O(\frac{y\log^2 y}{\log\log y})$

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

$O(\sqrt{\log y})$

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

$[0,\frac{\pi^2}{6}-1)\cup[1,\frac{\pi^2}{6}).$

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:

$x=\frac{1}{a_1}+\frac{1}{a_1a_2}+\frac{1}{a_1a_2a_3}+\cdots.$

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.

[edit] Open problems

For more information on this subject, see odd greedy expansion and Erdős–Straus conjecture.

Some notable problems remain unsolved with regard to Egyptian fractions, despite considerable effort by mathematicians.

The Erdős–Straus conjecture concerns the length of the shortest expansion for a fraction of the form 4/n. Does an expansion

$\frac4n=\frac1x+\frac1y+\frac1z$

exist for every n? It is known to be true for all n < 10¹⁴, and for all but a vanishingly small fraction of possible values of n, but the general truth of the conjecture remains unknown.

It is unknown whether an odd greedy expansion exists for every fraction with an odd denominator. If we modify Fibonacci's greedy method so that it always chooses the smallest possible odd denominator, under what conditions does this modified algorithm produce a finite expansion? An obvious necessary condition is that the starting fraction x/y have an odd denominator y, and it is conjectured but not known that this is also a sufficient condition. It is known (Breusch 1954; Stewart 1954) that every x/y with odd y has an expansion into distinct odd unit fractions, constructed using a different method than the greedy algorithm.

It is possible to use brute-force search algorithms to find the Egyptian fraction representation of a given number with the fewest possible terms (Stewart 1992) or minimizing the largest denominator; however, such algorithms can be quite inefficient. The existence of polynomial time algorithms for these problems, or more generally the computational complexity of such problems, remains unknown.

[edit] Bibliography

Beeckmans, L. (1993). "The splitting algorithm for Egyptian fractions". Journal of Number Theory 43: 173–185. DOI:10.1006/jnth.1993.1015. MR 1207497.

Botts, Truman (1967). "A chain reaction process in number theory". Mathematics Magazine: 55–65. MR 0209217.

Breusch, R. (1954). "A special case of Egyptian fractions, solution to advanced problem 4512". American Mathematical Monthly 61: 200–201.

Bruins, Evert M. (1957). "Platon et la table égyptienne 2/n". Janus 46: 253–263.

Eves, Howard (1953). An Introduction to the History of Mathematics. Holt, Reinhard, and Winston. ISBN 0-03-029558-0.

Gardner, Milo (2002). "The Egyptian Mathematical Leather Roll, attested short term and long term". Ivor Gratton-Guiness (ed.) History of the Mathematical Sciences: 119–134, Hindustan Book Co. ISBN 81-85931-45-3.

Gillings, Richard J. (1982). Mathematics in the Time of the Pharaohs. Dover. ISBN 0-486-24315-X.

Graham, R. L. (1964). "On finite sums of reciprocals of distinct nth powers". Pacific Journal of Mathematics 14 (1): 85–92. MR 0159788.

Hultsch, Friedrich (1895). Die Elemente der ägyptischen Theilungsrechnung. Leipzig: S. Hirzel.

Knorr, Wilbur R. (1982). "Techniques of fractions in ancient Egypt and Greece". Historia Mathematica 9: 133–171. DOI:10.1016/0315-0860(82)90001-5. MR 0662138.

Martin, G. (1999). "Dense Egyptian fractions". Transactions of the American Mathematical Society 351: 3641–3657. DOI:10.1090/S0002-9947-99-02327-2. MR 1608486.

Robins, Gay; Shute, Charles (1990). The Rhind Mathematical Papyrus: An Ancient Egyptian Text. Dover. ISBN 0-486-26407-6.

Sigler, Laurence E. (trans.) (2002). Fibonacci's Liber Abaci. Springer-Verlag. ISBN 0-387-95419-8.

Stewart, B. M. (1954). "Sums of distinct divisors". American Journal of Mathematics 76: 779–785. MR 0064800.

Stewart, I. (1992). "The riddle of the vanishing camel". Scientific American (June): 122–124.

Struik, Dirk J. (1967). A Concise History of Mathematics. Dover, 20–25. ISBN 0-486-60255-9.

Takenouchi, T. (1921). "On an indeterminate equation". Proceedings of the Physico-Mathematical Society of Japan, 3rd ser. 3: 78–92.

Tenenbaum, G.; Yokota, H. (1990). "Length and denominators of Egyptian fractions". Journal of Number Theory 35: 150–156. DOI:10.1016/0022-314X(90)90109-5. MR 1057319.