{"paper":{"title":"Long Intervals Without Distinct Multiples of the First $n$ Positive Integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Scott Duke Kominers","submitted_at":"2026-07-11T18:25:27Z","abstract_excerpt":"For positive integers $n$ and $m$, let $f(n,m)$ be the least integer $h\\ge0$ such that $(m,m+h]$ contains distinct integers $a_1,\\ldots,a_n$ satisfying $i\\mid a_i$ for $1\\le i\\le n$, and put $F(n)=\\max_{m\\in\\mathbb{N}} f(n,m)$. A recent theorem of van Doorn [INTEGERS, 2026; arXiv:2601.16972] gives $F(n)-f(n,n)>0.36\\,n\\log n/\\log\\log n$ for sufficiently large $n$. We prove \\[\n  \\liminf_{n\\to\\infty}\n  \\frac{F(n)-f(n,n)}{n\\log n}\n  \\ge \\frac{1}{\\mathrm{e}}. \\] Thus, for every fixed $c<1/\\mathrm{e}$ and all sufficiently large $n$, some interval of length $c\\,n\\log n$ contains no system of pairwise"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10431","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/2607.10431/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"}