2010/10/11 by Manuel Bello-Hernández, Bello-Hernández, Manuel, Manuel Benito +3
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1010.2035
24 pages. Summit to a journal. This is a new version of our old paper. We include new results, see the new abstract
openalex publication_date 2010/10/11 · arxiv created 2012/04/30 · arxiv updated 2012/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 1014. 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≤ n≤ 2× 1014, satisfy the Edős-Straus Conjecture.