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
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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

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Cite this paper

APA

Vassilev, P., and Vassilev-Missana, M. (2006). On some solved and unsolved problems related to Egyptian unit fractions. Notes on Number Theory and Discrete Mathematics, 12(2), 1-7.

Chicago

Vassilev, Peter, and Mladen Vassilev-Missana. “On Some Solved and Unsolved Problems Related to Egyptian Unit Fractions.” Notes on Number Theory and Discrete Mathematics 12, no. 2 (2006): 1-7.

MLA

Vassilev, Peter, and Mladen Vassilev-Missana. “On Some Solved and Unsolved Problems Related to Egyptian Unit Fractions.” Notes on Number Theory and Discrete Mathematics 12.2 (2006): 1-7. Print.

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