{"paper":{"title":"Counting the number of solutions to the Erdos-Straus equation on unit fractions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christian Elsholtz, Terence Tao","submitted_at":"2011-07-06T01:02:36Z","abstract_excerpt":"For any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $\\frac{4}{n} = \\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z}$ with $x,y,z$ positive integers. The \\emph{Erd\\H{o}s-Straus conjecture} asserts that $f(n) > 0$ for every $n \\geq 2$. To solve this conjecture, it suffices without loss of generality to consider the case when $n$ is a prime $p$.\n  In this paper we consider the question of bounding the sum $\\sum_{p<N} f(p)$ asymptotically as $N \\to \\infty$, where $p$ ranges over primes. Our main result establishes the asymptotic upper and lower bounds $$ N \\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.1010","kind":"arxiv","version":6},"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"}