  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 $1\le r\le
12l$, if at least one of the sums in right side of the expression, say,
$a+(4r-a-1),~1\le a\le 2r-1$ for at least one of the possible value of $r$ 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 $l=10^5$.
