# 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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## Sums of $k$ unit fractionsHTML articles powered by AMS MathViewer

by Christian Elsholtz
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 $k=3$.
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