{"paper":{"title":"Congruence Classes of Supporting the Erd\\\"{o}s-Straus Conjecture I: Tame Solutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xiaoping Xu","submitted_at":"2026-05-22T13:11:06Z","abstract_excerpt":"In 1948, Erd\\\"{o}s and Straus formulated a conjecture : for any positive integer $n>2$, there exist positive integers $n_1,n_2$ and $n_3$ such that \\begin{equation}\\frac{4}{n}=\\frac{1}{n_1}+\\frac{1}{n_2}+\\frac{1}{n_3},\\nonumber\\end{equation} which is still open. It is known that one only needs to prove the conjecture for any prime number $n$ such that $n\\equiv 1\\;(\\mbox{mod}\\;24)$. If $n=24m+1$ and $n_1\\leq n_2,n_3$, then $n_1=6m+k$ with $1\\leq k\\leq 12m$. A solution $(n_1,n_2,n_3)$ of the above equation is called a {\\it tame solution} if $n_2$ and $n_3$ are factors of $(6m+k)(24m+1)$. We call"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.23601","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.23601/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}