On Egyptian fractions

Abstract

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.