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In this paper, we propose a so-called weak residue conjecture(i.e., $\\frac{2^{E\\left( N \\right)}}{3^{O\\left( N \\right)}\\cdot N}\\le 2$). We prove that if 3x+1 conjecture is true and the weak residue conjecture is true, there exist non-trivial relationships among $D\\left( "},"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":"2506.23070","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2025-06-29T03:15:50Z","cross_cats_sorted":[],"title_canon_sha256":"c35321b51bb3ac70fc9c8ae433c7b96a07b45cb66ce2a85faca9be39095d1d56","abstract_canon_sha256":"6d07ebd7dfff2e3358d3dfd6daa3c154db0ea88265b65477ba51fa38f204f9b9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:35:14.659831Z","signature_b64":"KkmDmnIzlgaZ3xplC7sgF3FGaXgPfPWuOkn2MFnXD5jWr0a77Pzygbagx/tg+OpY2kZ+dKvEWUc6mnZhINM2DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"084820a0113058ceb2d58cdd44b798f20fecbf5e6168a422f3847a98baab99f9","last_reissued_at":"2026-07-05T11:35:14.659359Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:35:14.659359Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Iteration Steps of 3x+1 Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Youchun Luo","submitted_at":"2025-06-29T03:15:50Z","abstract_excerpt":"On the 3x+1 problem, given a positive integer $N$, let $D\\left( N \\right) $, $O\\left( N \\right) $, $E\\left( N \\right) $ be the total iteration steps, the odd iteration steps and the even iteration steps when $N$ iterates to 1(except 1) respectively. Trivially, we have $D\\left( N \\right) =O\\left( N \\right) +E\\left( N \\right) $. In this paper, we propose a so-called weak residue conjecture(i.e., $\\frac{2^{E\\left( N \\right)}}{3^{O\\left( N \\right)}\\cdot N}\\le 2$). We prove that if 3x+1 conjecture is true and the weak residue conjecture is true, there exist non-trivial relationships among $D\\left( "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.23070","kind":"arxiv","version":2},"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/2506.23070/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":"2506.23070","created_at":"2026-07-05T11:35:14.659413+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.23070v2","created_at":"2026-07-05T11:35:14.659413+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.23070","created_at":"2026-07-05T11:35:14.659413+00:00"},{"alias_kind":"pith_short_12","alias_value":"BBECBIARGBMM","created_at":"2026-07-05T11:35:14.659413+00:00"},{"alias_kind":"pith_short_16","alias_value":"BBECBIARGBMM5MWV","created_at":"2026-07-05T11:35:14.659413+00:00"},{"alias_kind":"pith_short_8","alias_value":"BBECBIAR","created_at":"2026-07-05T11:35:14.659413+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/BBECBIARGBMM5MWVRTOUJN4Y6I","json":"https://pith.science/pith/BBECBIARGBMM5MWVRTOUJN4Y6I.json","graph_json":"https://pith.science/api/pith-number/BBECBIARGBMM5MWVRTOUJN4Y6I/graph.json","events_json":"https://pith.science/api/pith-number/BBECBIARGBMM5MWVRTOUJN4Y6I/events.json","paper":"https://pith.science/paper/BBECBIAR"},"agent_actions":{"view_html":"https://pith.science/pith/BBECBIARGBMM5MWVRTOUJN4Y6I","download_json":"https://pith.science/pith/BBECBIARGBMM5MWVRTOUJN4Y6I.json","view_paper":"https://pith.science/paper/BBECBIAR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.23070&json=true","fetch_graph":"https://pith.science/api/pith-number/BBECBIARGBMM5MWVRTOUJN4Y6I/graph.json","fetch_events":"https://pith.science/api/pith-number/BBECBIARGBMM5MWVRTOUJN4Y6I/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BBECBIARGBMM5MWVRTOUJN4Y6I/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BBECBIARGBMM5MWVRTOUJN4Y6I/action/storage_attestation","attest_author":"https://pith.science/pith/BBECBIARGBMM5MWVRTOUJN4Y6I/action/author_attestation","sign_citation":"https://pith.science/pith/BBECBIARGBMM5MWVRTOUJN4Y6I/action/citation_signature","submit_replication":"https://pith.science/pith/BBECBIARGBMM5MWVRTOUJN4Y6I/action/replication_record"}},"created_at":"2026-07-05T11:35:14.659413+00:00","updated_at":"2026-07-05T11:35:14.659413+00:00"}