On Egyptian fractions

TL;DR

This paper introduces a polynomial that generates values satisfying the Erdős-Straus Conjecture for many numbers, proves the existence of arbitrarily long sequences of such numbers, and provides an algorithm to verify the conjecture up to 2×10^{14}.

Contribution

It presents a novel polynomial related to the conjecture, proves the existence of long sequences satisfying it, and offers an algorithm with proven convergence for verifying the conjecture for large ranges.

Findings

Polynomial values satisfy the Erdős-Straus Conjecture for many numbers.

Arbitrarily long sequences of consecutive numbers satisfy the conjecture.

Algorithm verifies the conjecture for all numbers up to 2×10^{14}.

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.