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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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by Christian Elsholtz PDF
Trans. Amer. Math. Soc. 353 (2001), 3209-3227 Request permission

Abstract:

Erdős 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 \[ E_{m,k}(N)= \mid \{ n \leq N \mid \frac {m}{n} = \frac {1}{t_1} + \ldots + \frac {1}{t_k} \text { has no solution with }t_i \in \mathbb {N} \} \mid . \] 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 \[ E_{m,k}(N) \ll N \exp \left ( -c_{m,k} (\log N)^{1-\frac {1}{2^{k-1}-1}} \right ) \text { with } c_{m,k}>0. \] 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 $k=3$.
References
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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
  • 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
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 3209-3227
  • MSC (2000): Primary 11D68; Secondary 11D72, 11N36
  • DOI: https://doi.org/10.1090/S0002-9947-01-02782-9
  • MathSciNet review: 1828604