{"paper":{"title":"A Note on Terence Tao's Paper \"On the Number of Solutions to 4/p=1/n_1+1/n_2+1/n_3\"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chaohua Jia","submitted_at":"2011-07-27T05:55:52Z","abstract_excerpt":"For the positive integer $n$, let $f(n)$ denote the number of positive integer solutions $(n_1,\\,n_2,\\,n_3)$ of the Diophantine equation $$ {4\\over n}={1\\over n_1}+{1\\over n_2}+{1\\over n_3}. $$ For the prime number $p$, $f(p)$ can be split into $f_1(p)+f_2(p),$ where $f_i(p)(i=1,\\,2)$ counts those solutions with exactly $i$ of denominators$n_1,\\,n_2,\\,n_3$ divisible by $p.$\n  Recently Terence Tao proved that $$ \\sum_{p< x}f_2(p)\\ll x\\log^2x\\log\\log x $$ with other results. But actually only the upper bound $x\\log^2x\\log\\log^2x$ can be obtained in his discussion. In this note we shall use an el"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.5394","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"}