On solutions of the Diophantine equation 1/x1+2/x2 +3/x3 + … + k/xk=1 when 2 ≤ x1 < x2 < x3 < … < xk are integers and k = x1

Nechemia Burshtein
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 2, Pages 30—35
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Nechemia Burshtein
117 Arlozorov Street
Tel Aviv 6209814, Israel

Abstract

Some general solutions of the title equation are established and exhibited. It is also shown that for each value of x1 when x1 ≤ 4, the equation has at least three solutions. For the particular values x1 = 2, 3, 4, all the solutions of the equation are determined and demonstrated.

Keywords

  • Diophantine equations
  • Egyptian fractions

AMS Classification

  • 11D68

References

  1. Stewart, B. M., & Webb, W. A. (1966) Sums of fractions with bounded numerators, Canadian J. Math., 18, 999–1003.
  2. Webb, W. A. (1976) On the Diophantine equation k/n=a1 /x1+a2 /x2+a3 /x3, Časopis pro Pĕstování Matematiky, 101, 360–365.

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Cite this paper

Burshtein, N. (2017). On solutions of the Diophantine equation 1/x1+2/x2 +3/x3 + … + k/xk=1 when 2 ≤ x1 < x2 < x3 < … < xk are integers and k = x1 . Notes on Number Theory and Discrete Mathematics, 23(2), 30-35.

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