Authors: Theophilus Agama
In this paper we discuss the Erd\'{o}s-Straus conjecture. Using a very simple method we show that for each $L\in \mathbb{N}$ with $L>n-1$ there exist some $(x_1,x_2,\ldots,x_n)\in \mathbb{N}^n$ with $x_i\neq x_j$ for all $1\leq i<j\leq n$ such that \begin{align}\frac{n}{L}\ll \sum \limits_{j=1}^{n}\frac{1}{x_j}\ll \frac{n}{L}\nonumber \end{align}In particular, for each $L\geq 3$ there exist some $(x_1,x_2,x_3)\in \mathbb{N}^3$ with $x_1\neq x_2$, $x_2\neq x_3$ and $x_3\neq x_1$ such that \begin{align}c_1\frac{3}{L}\leq \frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}\leq c_2\frac{3}{L}\nonumber \end{align}for some $c_1,c_2>1$.
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[v1] 2019-12-31 15:11:41
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