{"paper":{"title":"Maximum GCD Among Pairs of Random Integers","license":"http://creativecommons.org/licenses/publicdomain/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"E. E. Pyle, R. W. R. Darling","submitted_at":"2009-11-13T20:58:51Z","abstract_excerpt":"Fix $\\alpha >0$, and sample $N$ integers uniformly at random from $\\{1,2,\\ldots ,\\lfloor e^{\\alpha N}\\rfloor \\}$. Given $\\eta >0$, the probability that the maximum of the pairwise GCDs lies between $N^{2-\\eta }$ and $N^{2+\\eta}$ converges to 1 as $N\\to \\infty $. More precise estimates are obtained. This is a Birthday Problem: two of the random integers are likely to share some prime factor of order $N^2/\\log [N]$. The proof generalizes to any arithmetical semigroup where a suitable form of the Prime Number Theorem is valid."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.2660","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"}