On rational fractions not expressible as a sum of three unit fractions

Simon Brown
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 20, 2014, Number 2, Pages 61—64
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Authors and affiliations

Simon Brown
The Deviot Institute, Deviot, Tasmania 7275, Australia and
School of Human Life Sciences, University of Tasmania,
Locked Bag 1320, Launceston, Tasmania 7250, Australia

Abstract

Of those fractions (a / b < 1) that can not be expressed as a sum of three unit fractions, many can be written in terms of three unit fractions if the smallest denominator is ⎣b / a ⎦ and the next largest denominator is < 0. General expressions are given for some specific classes of these. Two examples of Yamamoto are reconsidered.

Keywords

  • Unit fraction

AMS Classification

  • 11D68

References

  1. Brown, S. Bounds of the denominators of Egyptian fractions, World Applied Programming, Vol. 2, 2012, 425‒430.
  2. Brown, S. On the number of sums of three unit fractions, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 4, 28‒32.
  3. Sierpinski, W. Sur les décompositions de nombres rationnels en fractions primaires, Mathesis, Vol. 65, 1956, 16‒32.
  4. Vaughan, R. C. On a problem of Erdös, Straus and Schinzel, Matematika, Vol. 17,
    1970, 193‒198.
  5. Vose, M. D. Egyptian fractions, Bulletin of the London Mathematical Society, Vol. 17, 1985, 21‒24.
  6. Webb, W. A. Rationals not expressible as a sum of three unit fractions, Elemente der Mathematik, Vol. 29, 1974, 1‒6.
  7. Yamamoto, K. On a conjecture of Erdös, Memoirs of the Faculty of Science, Kyushu University, Vol. 18, 1964, 166‒167.

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Cite this paper

APA

Brown, S. (2014). On rational fractions not expressible as a sum of three unit fractions. Notes on Number Theory and Discrete Mathematics, 20(2), 61-64.

Chicago

Brown, Simon. “On Rational Fractions Not Expressible as a Sum of Three Unit Fractions.” Notes on Number Theory and Discrete Mathematics 20, no. 2 (2014): 61-64.

MLA

Brown, Simon. “On Rational Fractions Not Expressible as a Sum of Three Unit Fractions.” Notes on Number Theory and Discrete Mathematics 20.2 (2014): 61-64. Print.

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