On some solved and unsolved problems related to Egyptian unit fractions

Peter Vassilev and Mladen Vassilev-Missana
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 12, 2006, Number 2, Pages 1–7
Full paper (PDF, 99 Kb)

Details

Authors and affiliations

Peter Vassilev
CLBME-Bulg. Academy of Sci., Sofia, Bulgaria

Mladen Vassilev-Missana
2. V. Hugo Str. 5, ap.3, Sofia-1124, Bulgaria

Abstract

In the paper a method named “Continuation of Unity” that is very useful for obtaining infinitely many decompositions of the number 1 with the help of Unit Egyptian fractions starting from given initial decomposition is demonstrated. The important question about decompositions of the number 1 with the help of unit fractions with distinct denominators that belong to a fixed increasing arithmetic progression of natural numbers is discussed. Also some applications related to fractions with odd denominators only are proposed.

Keywords

• Egyptian fraction
• Unit fraction
• Abundant number
• Pseudoperfect number

References

1. Vassilev, M., Two Theorems Concerning Divisors, Bulletin of Number Theory and Related Topics, Vol. 12, 1988, 10-19
2. Stewart,B., Sums of Distinct Divisors, American Journal of Mathematics, Vol. 76,
   1954: 779-785
3. http://eom.springer.deAa130070.htm
4. Dickson, E., History of the theory of numbers, I (Divisibility and primality), 1919