  We find a polynomial in three variables whose values at nonnegative integers
satisfy the Erd\H{o}s-Straus Conjecture. Although the perfect squares are not
covered by these values, it allows us to prove that there are arbitrarily long
sequence of consecutive numbers satisfying the Erd\H{o}s-Straus Conjecture. We
conjecture that the values of this polynomial include all the prime numbers of
the form $4q+5$, which is checked up to $10^{14}$. A greedy-type algorithm to
find an Erd\H{o}s-Straus decomposition is also given; the convergence of this
algorithm is proved for a wide class of numbers. Combining this algorithm with
the mentioned polynomial we verify that all the natural numbers $n$, $2\le n\le
2\times 10^{14}$, satisfy the Ed\H{o}s-Straus Conjecture.
