{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:ZTLATOIYXKNZGWSCYDZUHAYZ45","short_pith_number":"pith:ZTLATOIY","schema_version":"1.0","canonical_sha256":"ccd609b918ba9b935a42c0f3438319e761e0d12af8e12d1d35a3010f94abe66d","source":{"kind":"arxiv","id":"2105.03416","version":2},"attestation_state":"computed","paper":{"title":"On partitions of integers with restrictions involving squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chao Huang, Zhi-Wei Sun","submitted_at":"2021-05-04T16:09:12Z","abstract_excerpt":"In this paper, we study partitions of positive integers with restrictions involving squares. We mainly establish the following two results (which were conjectured by Sun in 2013):\n  (i) Each positive integer $n$ can be written as $n=x+y+z$ with $x,y,z$ positive integers such that $x^2+y^2+z^2$ is a square, unless $n$ has the form $n=2^{a}3^{b}$ or $2^{a}7$ with $a$ and $b$ nonnegative integers.\n  (ii) Each integer $n>7$ with $n\\not=11,14,17$ can be written as $n=x+y+2z$ with $x,y,z$ positive integers such that $x^2+y^2+2z^2$ is a square."},"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":"2105.03416","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-05-04T16:09:12Z","cross_cats_sorted":[],"title_canon_sha256":"1d611c36a4c6e298d2fa955ec1a4623f1303695802a13b1b0163f3f70cdc4c57","abstract_canon_sha256":"9ae22fbfcb2fb90828ae406e0cfb5ee23c73a7117a09d3a269b45a897af01009"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:43:34.377084Z","signature_b64":"fvE0mWGwVfS9iAy7pAringHTz+NV9VNFPq2ki42SKeA3v6XLp4Je7u4mqR1DGIbIcUaZaHuyiHNtOd31G/WgDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ccd609b918ba9b935a42c0f3438319e761e0d12af8e12d1d35a3010f94abe66d","last_reissued_at":"2026-07-05T02:43:34.376655Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:43:34.376655Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On partitions of integers with restrictions involving squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chao Huang, Zhi-Wei Sun","submitted_at":"2021-05-04T16:09:12Z","abstract_excerpt":"In this paper, we study partitions of positive integers with restrictions involving squares. We mainly establish the following two results (which were conjectured by Sun in 2013):\n  (i) Each positive integer $n$ can be written as $n=x+y+z$ with $x,y,z$ positive integers such that $x^2+y^2+z^2$ is a square, unless $n$ has the form $n=2^{a}3^{b}$ or $2^{a}7$ with $a$ and $b$ nonnegative integers.\n  (ii) Each integer $n>7$ with $n\\not=11,14,17$ can be written as $n=x+y+2z$ with $x,y,z$ positive integers such that $x^2+y^2+2z^2$ is a square."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.03416","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/2105.03416/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":"2105.03416","created_at":"2026-07-05T02:43:34.376713+00:00"},{"alias_kind":"arxiv_version","alias_value":"2105.03416v2","created_at":"2026-07-05T02:43:34.376713+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2105.03416","created_at":"2026-07-05T02:43:34.376713+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZTLATOIYXKNZ","created_at":"2026-07-05T02:43:34.376713+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZTLATOIYXKNZGWSC","created_at":"2026-07-05T02:43:34.376713+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZTLATOIY","created_at":"2026-07-05T02:43:34.376713+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/ZTLATOIYXKNZGWSCYDZUHAYZ45","json":"https://pith.science/pith/ZTLATOIYXKNZGWSCYDZUHAYZ45.json","graph_json":"https://pith.science/api/pith-number/ZTLATOIYXKNZGWSCYDZUHAYZ45/graph.json","events_json":"https://pith.science/api/pith-number/ZTLATOIYXKNZGWSCYDZUHAYZ45/events.json","paper":"https://pith.science/paper/ZTLATOIY"},"agent_actions":{"view_html":"https://pith.science/pith/ZTLATOIYXKNZGWSCYDZUHAYZ45","download_json":"https://pith.science/pith/ZTLATOIYXKNZGWSCYDZUHAYZ45.json","view_paper":"https://pith.science/paper/ZTLATOIY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2105.03416&json=true","fetch_graph":"https://pith.science/api/pith-number/ZTLATOIYXKNZGWSCYDZUHAYZ45/graph.json","fetch_events":"https://pith.science/api/pith-number/ZTLATOIYXKNZGWSCYDZUHAYZ45/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZTLATOIYXKNZGWSCYDZUHAYZ45/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZTLATOIYXKNZGWSCYDZUHAYZ45/action/storage_attestation","attest_author":"https://pith.science/pith/ZTLATOIYXKNZGWSCYDZUHAYZ45/action/author_attestation","sign_citation":"https://pith.science/pith/ZTLATOIYXKNZGWSCYDZUHAYZ45/action/citation_signature","submit_replication":"https://pith.science/pith/ZTLATOIYXKNZGWSCYDZUHAYZ45/action/replication_record"}},"created_at":"2026-07-05T02:43:34.376713+00:00","updated_at":"2026-07-05T02:43:34.376713+00:00"}