{"paper":{"title":"Denominators of Bernoulli polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Florian Luca, Igor E. Shparlinski, Olivier Bordell\\`es, Pieter Moree","submitted_at":"2017-06-29T15:27:56Z","abstract_excerpt":"For a positive integer $n$ let $\\mathfrak{P}_n=\\prod_{s_p(n)\\ge p} p,$ where $p$ runs over all primes and $s_p(n)$ is the sum of the base $p$ digits of $n$. For all $n$ we prove that $\\mathfrak{P}_n$ is divisible by all \"small\" primes with at most one exception. We also show that $\\mathfrak{P}_n$ is large, has many prime factors exceeding $\\sqrt{n}$, with the largest one exceeding $n^{20/37}$. We establish Kellner's conjecture, which says that the number of prime factors exceeding $\\sqrt{n}$ grows asymptotically as $\\kappa \\sqrt{n}/\\log n$ for some constant $\\kappa$ with $\\kappa=2$. Further, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.09804","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"}