{"paper":{"title":"A Greatest Common Divisor Criterion of Certain Binomial Coefficients","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LO"],"primary_cat":"math.NT","authors_text":"Dakai Guo, Ruichen Qiu, Ruyong Feng, Xiao-Shan Gao, Yichuan Cao","submitted_at":"2026-06-22T08:14:01Z","abstract_excerpt":"The binomial greatest common divisor (gcd) criterion recorded as OEIS A080170 is proven. The criterion also appears as conjecture (17) in Ralf Stephan's list of OEIS conjectures. For $k\\geq 2$, put \\[\n  D(k)=\\gcd_{2\\leq q\\leq k+1}\\binom{qk}{k},\n  \\qquad n=k+1. \\] If $P$ is the largest prime-power component $p^a$ exactly dividing $n$, then the criterion asserts \\[\n  D(k)=1 \\quad\\Longleftrightarrow\\quad \\frac{n}{P}>P. \\] The proof is formalized in Lean and the Lean artifact is accepted as part of the Formal Conjectures project. Both the natural-language proof and the Lean formalization are gener"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22997","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/2606.22997/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"}