{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:AFIRTQCTCLICCKNOZ3HGSKM4C5","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"783383476f29efcaa74abe31ae4b45499dc598043a2cff3b27202c924602dd8b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-27T13:17:09Z","title_canon_sha256":"93c294cb1e52e027815e45b3829c9487e80b7c2adc16743456ecc89517905aec"},"schema_version":"1.0","source":{"id":"2605.28449","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.28449","created_at":"2026-05-28T02:04:53Z"},{"alias_kind":"arxiv_version","alias_value":"2605.28449v1","created_at":"2026-05-28T02:04:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.28449","created_at":"2026-05-28T02:04:53Z"},{"alias_kind":"pith_short_12","alias_value":"AFIRTQCTCLIC","created_at":"2026-05-28T02:04:53Z"},{"alias_kind":"pith_short_16","alias_value":"AFIRTQCTCLICCKNO","created_at":"2026-05-28T02:04:53Z"},{"alias_kind":"pith_short_8","alias_value":"AFIRTQCT","created_at":"2026-05-28T02:04:53Z"}],"graph_snapshots":[{"event_id":"sha256:1ed8ef23303fec96145ff56d3d4bf97da606665243694c6d80d021cdf55c203c","target":"graph","created_at":"2026-05-28T02:04:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2605.28449/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $C_n=n2^n+1$ denote the $n$th Cullen number. There has been recent interest in finding all Cullen numbers having a given Diophantine property. 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 $2,3,5,7$.","authors_text":"Divyum Sharma, Vikas Godara","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-27T13:17:09Z","title":"Additive Diophantine Equations involving S-Units, Factorials and Ternary Recurrences with repeated root"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.28449","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d0153324eca3cd4db7be11499270ba41d3ce99ae8bf856c9a8e7294a4a527e1e","target":"record","created_at":"2026-05-28T02:04:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"783383476f29efcaa74abe31ae4b45499dc598043a2cff3b27202c924602dd8b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-27T13:17:09Z","title_canon_sha256":"93c294cb1e52e027815e45b3829c9487e80b7c2adc16743456ecc89517905aec"},"schema_version":"1.0","source":{"id":"2605.28449","kind":"arxiv","version":1}},"canonical_sha256":"015119c05312d02129aecece69299c1768dfa4748aa6cc8549d546176e87d157","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"015119c05312d02129aecece69299c1768dfa4748aa6cc8549d546176e87d157","first_computed_at":"2026-05-28T02:04:53.378044Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-28T02:04:53.378044Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iU6DIh2DOjE2rsiIo9IbslJzmF2Ndfkie/SczlDkKVTjAPYNu0kVnrQ+LQuQzXVB0GSwABSf40dwWL5Nz5NlAw==","signature_status":"signed_v1","signed_at":"2026-05-28T02:04:53.378584Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.28449","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d0153324eca3cd4db7be11499270ba41d3ce99ae8bf856c9a8e7294a4a527e1e","sha256:1ed8ef23303fec96145ff56d3d4bf97da606665243694c6d80d021cdf55c203c"],"state_sha256":"2cf421451d576c3352c0226bc00b3a05cf43547c672163703bf8787ff8109663"}