On Egyptian fractions

Abstract

We find a polynomial in three variables whose values at nonnegative integers satisfy the Erdős-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 Erdős-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 Erdős-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 Edős-Straus Conjecture.

arXiv comment24 pages. Summit to a journal. This is a new version of our old paper. We include new results, see the new abstract

References in corpus (1)

• Counting the number of solutions to the Erdos-Straus equation on unit fractions