Egyptian fraction

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Egyptian fraction

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Egyptian fraction (plural Egyptian fractions)

(number theory) A representation of a rational number as a sum of distinct unit fractions.
The fraction ${\frac {7}{10}}$ can be written as the Egyptian fraction ${\frac {1}{2}}+{\frac {1}{5}}$ .

The calculation of $M^{N}$ where both M and N are real numbers can be done by expressing N as the sum of an integer h and an Egyptian fraction $1/g_{1}+1/g_{2}+1/g_{3}+...$ . Then $M^{N}=M^{h}{\sqrt[{g_{1}}]{M}}{\sqrt[{g_{2}}]{M}}{\sqrt[{g_{3}}]{M}}...$ where each of the roots can be calculated by means of the Newton–Raphson method.
- 1995, C. Pomerance, A. Sárkőzy, Combinatorial number theory, R. L. Graham, M. Grötschel, L. Lovász (edtors), Handbook of Combinatorics, Elsevier (North-Holland), page 1014,
  However, a representation of r as a sum of distinct Egyptian fractions is certainly not unique and this fact leads to many questions.
- 2013, R. L. Graham, “Paul Erdős and Egyptian Fractions”, in László Lovász, Imre Ruzsa, Vera T. Sós, editors, Erdős Centennial, Springer,, page 293:
  In [27], Erdős also considers various questions relating to Egyptian fraction decompositions of $\textstyle 1=\sum _{k=1}^{n}{\frac {1}{x_{k}}}$ .
- 2016, Annette Imhausen, Mathematics in Ancient Egypt: A Contextual History, Princeton University Press, page 5:
  The modern description of Egyptian fraction reckoning being "restricted" to unit fractions is obviously anachronistic (indeed, the Egyptian concept of fractions did not include a numerator, but from a historian's point of view this cannot be criticized on the basis that our modern fractions consist of denominator and numerator). Furthermore, this criticism does not do justice to the development of Egyptian fractions.
(number theory, rare) A unit fraction.