Asymptotic estimates on the Erd\H{o}s-Straus conjecture

Abstract

In this paper analyzes \textit{The Erd\H{o}s-Straus conjecture} asserts that $f$$(n)$ $>$ 0 for every $n$ $\geq$ 2, where $f(n)$ indicates the number of solutions to the Diophantine Equation $\frac{4}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}$. We show that there exists a function $G(p)$ to be a boundary asymptotic of $\sum_{p\leq{N}}f_{I}(p)$, which will have an associated error. We analyze the case when n is a prime number, this was separately developed by Terence Tao [8] and Jia [1], [2].