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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sums of $k$ unit fractions


Author: Christian Elsholtz
Journal: Trans. Amer. Math. Soc. 353 (2001), 3209-3227
MSC (2000): Primary 11D68; Secondary 11D72, 11N36
Published electronically: April 12, 2001
MathSciNet review: 1828604
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Abstract:

Erdos and Straus conjectured that for any positive integer $n\geq 2$ the equation $\frac{4}{n}= \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$has a solution in positive integers $x,\, y$, and $z$. Let $m > k \geq 3$ and

\begin{displaymath}E_{m,k}(N)= \, \mid \{ n \leq N \mid \frac{m}{n} = \frac{1}... ...t_k} \text{ has no solution with }t_i \in \mathbb{N}\} \mid . \end{displaymath}

We show that parametric solutions can be used to find upper bounds on $E_{m,k}(N)$where the number of parameters increases exponentially with $k$. This enables us to prove

\begin{displaymath}E_{m,k}(N) \ll N \exp \left( -c_{m,k} (\log N)^{1-\frac{1}{2^{k-1}-1}} \right) \text{ with } c_{m,k}>0. \end{displaymath}

This improves upon earlier work by Viola (1973) and Shen (1986), and is an ``exponential generalization'' of the work of Vaughan (1970), who considered the case $k=3$.


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Additional Information

Christian Elsholtz
Affiliation: Institut für Mathematik, Technische Universität Clausthal, Erzstrasse 1, D-38678 Clausthal-Zellerfeld, Germany
Email: elsholtz@math.tu-clausthal.de

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02782-9
PII: S 0002-9947(01)02782-9
Received by editor(s): May 23, 2000
Received by editor(s) in revised form: August 28, 2000
Published electronically: April 12, 2001
Additional Notes: The research for this paper was supported by a Ph.D.\ grant from the German National Merit Foundation
Article copyright: © Copyright 2001 American Mathematical Society