  The Erd\"{o}s-Straus conjecture states that the equation
$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ has positive integer
solutions $x, y, z$ for every postive integers $n\ge 2$. We generalize the
Erd\"{o}s-Straus equation, state several methods for obtaining well-defined
equations, and some features of well-defined equations. Using well-defined
equations, we explain the inefficiency of the solutions that did not lead to a
complete proof of the problem. Finally, we prove that this conjecture cannot be
completely proven by the methods expressed in various articles. For this
reason, we express conjectures equivalent to the Erd\"{o}s-Straus conjecture
and make the concept of conjecture clearer. The well-defined equations of
Erd\"{o}s-Straus have a lot to do with the concept of the least common multiple
and the greatest common divisor, and in the last section, we will independently
express the functions and features based on the subject of the least common
multiple.
