{"paper":{"title":"On Egyptian fractions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Emilio Fern\\'andez, Manuel Bello-Hern\\'andez, Manuel Benito","submitted_at":"2010-10-11T08:28:36Z","abstract_excerpt":"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. Com"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.2035","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}