{"paper":{"title":"Share at least half the numbers in a nontrivial LCM-closed set a nontrivial divisor?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tom Fischer","submitted_at":"2018-08-28T12:22:56Z","abstract_excerpt":"For a finite set of non-zero natural numbers that contains at least one element different from 1 and the least common multiple of any of its subsets, there exists a subset of at least half of its members which has a common divisor larger than 1. Utilizing a representation of the natural numbers as an order-theoretical ring of prime power sets, this conjecture is shown to be equivalent to Frankl's union-closed sets conjecture. Some results for cases where the conjecture, which also has meaningful interpretations in graph and lattice theory, is known to hold are provided. An equivalent dual vers"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.09247","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"}