Abstract:

In this paper, we are going to investigate the ${\bf{\textit{Erdős-Straus Conjecture }}}$: For any positive $n \geq 2$, there exists positive integers $k,k_1,k_2$ such that

$$ \dfrac{4}{n} = \dfrac{1}{k}+\dfrac{1}{k_1}+\dfrac{1}{k_2}$$

Firstly, we will solve a simpler form $ \dfrac{3}{n} = \dfrac{1}{x} + \dfrac{1}{y}$ as a starting point. Next we will investigate the Erdős-Straus Conjecture in the following dimensions: the related geometric representation of the Erdös-Straus Conjecture, the properties of solutions of the Erdős-Straus Conjecture, further investigation of some paper of the Erdös-Straus Conjecture, existence of special forms of solutions of the Erdős-Straus Conjecture, and the investigation of the Erdős-Straus Conjecture in algebraic dimension. The aim of this report is to find evidence that shows the ${\bf{\textit{Erdős-Straus Conjecture}}}$ is true. If evidence is not strong enough, we still hope that this report can make an improvement to the researched result at present.

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