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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1706.09804","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-06-29T15:27:56Z","cross_cats_sorted":[],"title_canon_sha256":"1a231d69092ed8291dd39df91a062f6af16a8174d474ff45ceb806fc036df1a7","abstract_canon_sha256":"19f9deec8af24cb4452185399b569612e916ccfa7e29096d60ca73f70e96b01c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:43.377269Z","signature_b64":"niXMMWexz5/7DH+kn4vpEfcOQbUbhQN/DLe2D03lbELZsqlsEU7/Cg/Nshi4SXVGZ8QwYC1+2oWgd4c73aj6DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c47790c0ea38bbd3ae7c4e8a8472700636f6c69a0b435428f37463f68186189c","last_reissued_at":"2026-05-18T00:17:43.376653Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:43.376653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\kappa=2$. 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