Tieling Chen and Reginald Koo

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 19, 2013, Number 2, Pages 15—25

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## 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

- 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. - Gay, R., C. Shute, The Rhind Mathematical Papyrus: an Ancient Egyptian Text, British Museum Press, London, 1987.
- Guy Richard K. Unsolved problems in number theory, 3rd ed., New York: Springer-Verlag, 2004.
- 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.

## Related papers

## Cite this paper

APAChen, T., & Koo, R. (2013). Two-term Egyptian fractions. Notes on Number Theory and Discrete Mathematics, 19(2), 15-25.

ChicagoChen, Tieling, and Reginald Koo. “Two-term Egyptian Fractions.” Notes on Number Theory and Discrete Mathematics 19, no. 2 (2013): 15-25.

MLAChen, Tieling, and Reginald Koo. “Two-term Egyptian Fractions.” Notes on Number Theory and Discrete Mathematics 19.2 (2013): 15-25. Print.