{"paper":{"title":"On the exceptional set for binary Egyptian fractions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jing-Jing Huang, Robert C. Vaughan","submitted_at":"2011-11-10T19:43:49Z","abstract_excerpt":"For fixed integer $a\\ge3$, we study the binary Diophantine equation $\\frac{a}n=\\frac1x+\\frac1y$ and in particular the number $E_a(N)$ of $n\\le N$ for which the equation has no positive integer solutions in $x, y$. The asymptotic formula $$E_a(N)\\sim C(a) \\frac{N(\\log\\log N)^{2^{m-1}-1}}{(\\log N)^{1-1/2^m}}$$ as $N$ goes to infinity, is established in this article, and this improves the best result in the literature dramatically. The proof depends on a very delicate analysis of the underlying group structure."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2574","kind":"arxiv","version":1},"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"}