{"paper":{"title":"The Sylow Divisor Condition: a Resolution of Erd\\H{o}s Problem 768","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Eric Li (Trinity College, University of Cambridge)","submitted_at":"2026-06-23T17:51:30Z","abstract_excerpt":"We resolve Erd\\H{o}s Problem 768. Let $A(x)$ count the positive integers $n\\le x$ such that, for every prime $p\\mid n$, there is a divisor $d>1$ of $n$ with $d\\equiv 1 \\pmod p$. Erd\\H{o}s asked whether $A(x)/x=\\exp(-(c+o(1))\\sqrt{\\log x}\\log\\log x)$ for some constant $c>0$. We prove that this holds with $c=1/(2\\sqrt{\\log 2})$; equivalently, $\\log(x/A(x))/(\\sqrt{\\log x}\\log\\log x)$ tends to $1/(2\\sqrt{\\log 2})$. The lower bound is obtained from primes in disjoint logarithmic intervals using a fourth-moment argument based on the multiplicative large sieve and a subset-product second moment. The "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24872","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/2606.24872/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"}