Two-term Egyptian fractions

Tieling Chen and Reginald Koo
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 19, 2013, Number 2, Pages 15–25
Full paper (PDF, 190 Kb)

Details

Authors and affiliations

Tieling Chen
Department of Mathematical Sciences
University of South Carolina Aiken, USA

Reginald Koo
Department of Mathematical Sciences
University of South Carolina Aiken, USA

Abstract

We study two-term Egyptian fraction representations of a given rational number. We consider the case of m/n where each prime factor p of n satisfies p ≡ ±1 (mod m): necessary and sufficient conditions for the existence of proper two-term Egyptian fraction expressions of such m/n are given, together with methods to find these representations. Furthermore, we determine the number of proper two-term Egyptian fraction expressions for 1/m, 2/m, 3/m, 4/m and 6/m.

Keywords

• Egyptian fractions
• Unit fraction
• Diophantine equation

AMS Classification

• 11D68

References

1. Bartoš, P. A remark on the number of solutions of the equation 1/x + 1/y = a/b in natural numbers, Časopis Pešt. Mat. Vol. 95, 1970, 411–415.
2. Gay, R., C. Shute, The Rhind Mathematical Papyrus: an Ancient Egyptian Text, British Museum Press, London, 1987.
3. Guy Richard K. Unsolved problems in number theory, 3rd ed., New York: Springer-Verlag, 2004.
4. Rav, Y. On the representation of rational numbers as a sum of a fixed number of unit fractions, J. Reine Angew. Math. Vol. 222, 1966, 207–213.