Unit Fraction

A unit fraction is a Fraction with Numerator 1, also known as an Egyptian Fraction. Any Rational Number has infinitely many representations as a sum of unit fractions, although for a given fixed number of terms, there are only finitely many. Each Fraction $x/y$ with $y$ Odd has a unit fraction representation in which each Denominator is Odd (Breusch 1954; Guy 1994, p. 160). Every $x/y$ has a $t$-term representation where $t={\mathcal O}(\sqrt{\log y}\,)$ (Vose 1985).

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.

See also Calcus, Half, Quarter, Scruple, Uncia

References

Beck, A.; Bleicher, M. N.; and Crowe, D. W. Excursions into Mathematics. New York: Worth Publishers, 1970.

Beeckmans, L. ``The Splitting Algorithm for Egyptian Fractions.'' J. Number Th. 43, 173-185, 1993.

Bleicher, M. N. ``A New Algorithm for the Expansion of Continued Fractions.'' J. Number Th. 4, 342-382, 1972.

Breusch, R. ``A Special Case of Egyptian Fractions.'' Solution to advanced problem 4512. Amer. Math. Monthly 61, 200-201, 1954.

Brown, K. S. ``Egyptian Unit Fractions.'' http://www.seanet.com/~ksbrown/.

Eppstein, D. ``Ten Algorithms for Egyptian Fractions.'' Math. Edu. Res. 4, 5-15, 1995.

Eppstein, D. ``Egyptian Fractions.'' http://www.ics.uci.edu/~eppstein/numth/egypt/.

Eppstein, D. Egypt.ma Mathematica notebook. http://www.ics.uci.edu/~eppstein/numth/egypt/egypt.ma.

Graham, R. ``On Finite Sums of Unit Fractions.'' Proc. London Math. Soc. 14, 193-207, 1964.

Guy, R. K. ``Egyptian Fractions.'' §D11 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 87-93 and 158-166, 1994.

Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory. Washington, DC: Math. Assoc. Amer., pp. 175-177 and 206-208, 1991.

Niven, I. and Zuckerman, H. S. An Introduction to the Theory of Numbers, 5th ed. New York: Wiley, p. 200, 1991.

Stewart, I. ``The Riddle of the Vanishing Camel.'' Sci. Amer., 122-124, June 1992.

Tenenbaum, G. and Yokota, H. ``Length and Denominators of Egyptian Fractions.'' J. Number Th. 35, 150-156, 1990.

Vose, M. ``Egyptian Fractions.'' Bull. London Math. Soc. 17, 21, 1985.

Wagon, S. ``Egyptian Fractions.'' §8.6 in Mathematica in Action. New York: W. H. Freeman, pp. 271-277, 1991.