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In particular, this means that all trajectories except for the trivial loop go through an element of $\\{3+4\\0\\}$ ($\\{5+8\\0\\}$ for the original mapping). I give reasons for this conjecture. Next, I note that the 3n+1 numbers and the 3n+3 numbers are the only numbers from the generalization $3n+p, p \\in \\{\\ldots,-3,-1,1,3,\\l"},"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":"1908.01509","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-05T08:16:41Z","cross_cats_sorted":[],"title_canon_sha256":"e3bf90f222a08a0595887c60a11e0b86dcba525a03a35c492de603e1cb770424","abstract_canon_sha256":"5d86aca07bbc900649ba8e9f1e5df7265d0e394b6f985e99a7308143ad2a1d3e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:51:31.131927Z","signature_b64":"5lCX1BCPgpAHElVuuIq63fc3p2RlYcqJqkQWoRGpgWvJbA8JXTfyan5LPQ6RPiZQiic41z7KATjsemsAMklhAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"478886b8eb86e0f9a24d9f23ac87e59fab3085cbd4e6453a230089ddc896ea7c","last_reissued_at":"2026-07-04T23:51:31.131585Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:51:31.131585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The 3n+1 problem: a partition of interest","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Maarten J. Wensink","submitted_at":"2019-08-05T08:16:41Z","abstract_excerpt":"A mapping conjugate to the Collatz mapping seems to imply that $\\N=\\{1,2,3,\\ldots\\}$ is partitioned in a trivial loop $\\{1\\}$ and `strings' that are ordered subsets of $\\{\\N \\setminus 1\\}$ that run from an element of $\\{2+3\\0\\}$ to an element of $\\{3+4\\0\\}$ ($\\0=0 \\cup \\N$). In particular, this means that all trajectories except for the trivial loop go through an element of $\\{3+4\\0\\}$ ($\\{5+8\\0\\}$ for the original mapping). I give reasons for this conjecture. Next, I note that the 3n+1 numbers and the 3n+3 numbers are the only numbers from the generalization $3n+p, p \\in \\{\\ldots,-3,-1,1,3,\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01509","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/1908.01509/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1908.01509","created_at":"2026-07-04T23:51:31.131643+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.01509v1","created_at":"2026-07-04T23:51:31.131643+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.01509","created_at":"2026-07-04T23:51:31.131643+00:00"},{"alias_kind":"pith_short_12","alias_value":"I6EINOHLQ3QP","created_at":"2026-07-04T23:51:31.131643+00:00"},{"alias_kind":"pith_short_16","alias_value":"I6EINOHLQ3QPTISN","created_at":"2026-07-04T23:51:31.131643+00:00"},{"alias_kind":"pith_short_8","alias_value":"I6EINOHL","created_at":"2026-07-04T23:51:31.131643+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/I6EINOHLQ3QPTISNT4R2ZB7FT6","json":"https://pith.science/pith/I6EINOHLQ3QPTISNT4R2ZB7FT6.json","graph_json":"https://pith.science/api/pith-number/I6EINOHLQ3QPTISNT4R2ZB7FT6/graph.json","events_json":"https://pith.science/api/pith-number/I6EINOHLQ3QPTISNT4R2ZB7FT6/events.json","paper":"https://pith.science/paper/I6EINOHL"},"agent_actions":{"view_html":"https://pith.science/pith/I6EINOHLQ3QPTISNT4R2ZB7FT6","download_json":"https://pith.science/pith/I6EINOHLQ3QPTISNT4R2ZB7FT6.json","view_paper":"https://pith.science/paper/I6EINOHL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.01509&json=true","fetch_graph":"https://pith.science/api/pith-number/I6EINOHLQ3QPTISNT4R2ZB7FT6/graph.json","fetch_events":"https://pith.science/api/pith-number/I6EINOHLQ3QPTISNT4R2ZB7FT6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I6EINOHLQ3QPTISNT4R2ZB7FT6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I6EINOHLQ3QPTISNT4R2ZB7FT6/action/storage_attestation","attest_author":"https://pith.science/pith/I6EINOHLQ3QPTISNT4R2ZB7FT6/action/author_attestation","sign_citation":"https://pith.science/pith/I6EINOHLQ3QPTISNT4R2ZB7FT6/action/citation_signature","submit_replication":"https://pith.science/pith/I6EINOHLQ3QPTISNT4R2ZB7FT6/action/replication_record"}},"created_at":"2026-07-04T23:51:31.131643+00:00","updated_at":"2026-07-04T23:51:31.131643+00:00"}