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We prove that, for a fixed integer $k$ and bounded integers $a_1,\\ldots,a_k$, the greatest prime divisor of $C_n-a_1m_1!-\\cdots-a_km_k!$ tends to infinity, in an effective way. We prove this for some more general families of ternary recurrence sequences as well. We also solve the Diophantine equation $$C_n = m_1! + m_2! + s,$$ where $s$ is a positive integer composed of primes $2,3,5,7$."},"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":"2605.28449","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-27T13:17:09Z","cross_cats_sorted":[],"title_canon_sha256":"93c294cb1e52e027815e45b3829c9487e80b7c2adc16743456ecc89517905aec","abstract_canon_sha256":"783383476f29efcaa74abe31ae4b45499dc598043a2cff3b27202c924602dd8b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-28T02:04:53.378584Z","signature_b64":"iU6DIh2DOjE2rsiIo9IbslJzmF2Ndfkie/SczlDkKVTjAPYNu0kVnrQ+LQuQzXVB0GSwABSf40dwWL5Nz5NlAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"015119c05312d02129aecece69299c1768dfa4748aa6cc8549d546176e87d157","last_reissued_at":"2026-05-28T02:04:53.378044Z","signature_status":"signed_v1","first_computed_at":"2026-05-28T02:04:53.378044Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Additive Diophantine Equations involving S-Units, Factorials and Ternary Recurrences with repeated root","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Divyum Sharma, Vikas Godara","submitted_at":"2026-05-27T13:17:09Z","abstract_excerpt":"Let $C_n=n2^n+1$ denote the $n$th Cullen number. 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