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

TL;DR

This paper investigates the asymptotic behavior of solutions to the Erdős-Straus conjecture, focusing on prime numbers and establishing bounds for the sum of solutions with an associated error term.

Contribution

It introduces a boundary asymptotic function for the sum of solutions over primes and analyzes the case when n is prime, extending previous work.

Findings

Existence of a boundary asymptotic function G(p) for the sum of solutions

Analysis of the case when n is prime, building on prior work by Tao and Jia

Establishment of an associated error term in the asymptotic estimate

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].