A Study on Erd\H{o}s-Straus conjecture on Diophantine equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$

The Erd\H{o}s-Straus conjecture is a renowned problem which describes that for every natural number $n(\ge 2)$, $\frac{4}{n}$ can be represented as the sum of three unit fractions. The main purpose of this study is to show that the Erd\H{o}s-Straus conjecture is true. The study also re-demonstrates Mordell theorem which states that $\frac{4}{n}$ has a expression as the sum of three unit fractions for every number $n$ except possibly for those primes of the form $n\equiv r$ (mod 780) with $r=1^2,{11}^2,{13}^2,{17}^2,{19}^2,{23}^2$. For $l,r,a\in \mathbb{N}$; $\frac{4}{24l+1}-\frac{1}{6l+r}=\frac{4r-1}{(6l+r)(24l+1)}$ with $1\le r\le 12l$,ifatleastoneofthesumsinrightsideoftheexpression,say,$a+(4r-a-1),~1\le a\le 2r-1$foratleastoneofthepossiblevalueof$r$ such that $a,(4r-a-1)$ divide $(6l+r)(24l+1)$; then the conjecture is valid for the corresponding $l$. However, in this way the conjecture can not be proved only twelve values of $l$ for $l$ up to $l={10}^5$.