{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:3MJN37X5RDWW7XHHAVQGQNXHQG","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":"70b8cd4a3cb181c7a94670df68f5e5845904c54f9ed35346006c6c19a4edbc20","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.NT","submitted_at":"2013-07-17T21:22:33Z","title_canon_sha256":"cefb82e56eded83774cc417c3da2e83506ca8179b4d2a30e96b945f3a745bd45"},"schema_version":"1.0","source":{"id":"1307.5328","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.5328","created_at":"2026-05-18T03:17:56Z"},{"alias_kind":"arxiv_version","alias_value":"1307.5328v1","created_at":"2026-05-18T03:17:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.5328","created_at":"2026-05-18T03:17:56Z"},{"alias_kind":"pith_short_12","alias_value":"3MJN37X5RDWW","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"3MJN37X5RDWW7XHH","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"3MJN37X5","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:4ace12b0d8397532cc2a798efc0e77ba23bf927ec8a5cbdda15cc9616a17f646","target":"graph","created_at":"2026-05-18T03:17:56Z","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"},"paper":{"abstract_excerpt":"In this work, we accomplish three goals. First, we determine the entire family of positive integer solutions to the three- variable Diophantine equation, xy=z^2; for n=2,3,4,5,6. For n=2, we obtain a 3-parameter family of solutions; for n=3, a 5-parameter of solutions; likewise for n=4. For n=5, a 7-parameter family of solutions; and likewise for n=6. See Theorems 2 through 6 respectively. The second goal of this paper, is determining all the positive integer solutions of xyz=w^2. This is done in Theorem7; the solution set is described in terms of six independent parameters. Finally, in Theore","authors_text":"Konstantine Zelator","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.NT","submitted_at":"2013-07-17T21:22:33Z","title":"The Diophantine equation xy=z^n; for n=2,3,4,5,6; the Diophantine equation xyz=w^2; and the Diophantine system: xy=v^2 and yz=w^2"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5328","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":"sha256:0704277e3cc0a84051b4bf68a683d4252a866ee1700f59d16281e61b5e9075a6","target":"record","created_at":"2026-05-18T03:17:56Z","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":"70b8cd4a3cb181c7a94670df68f5e5845904c54f9ed35346006c6c19a4edbc20","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.NT","submitted_at":"2013-07-17T21:22:33Z","title_canon_sha256":"cefb82e56eded83774cc417c3da2e83506ca8179b4d2a30e96b945f3a745bd45"},"schema_version":"1.0","source":{"id":"1307.5328","kind":"arxiv","version":1}},"canonical_sha256":"db12ddfefd88ed6fdce705606836e781860f9a65527aa49fcad80429844415da","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"db12ddfefd88ed6fdce705606836e781860f9a65527aa49fcad80429844415da","first_computed_at":"2026-05-18T03:17:56.091891Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:17:56.091891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"x9AylY+qFTfEysvRkvTa4bTdxxWjHEB1ryKeZHuE1wrmTTa+/xMT2nSv0UoOLbsGYqf86YlVJH+bStObB7YeCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:17:56.092620Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.5328","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0704277e3cc0a84051b4bf68a683d4252a866ee1700f59d16281e61b5e9075a6","sha256:4ace12b0d8397532cc2a798efc0e77ba23bf927ec8a5cbdda15cc9616a17f646"],"state_sha256":"1e70104c1a8e5e09a1bcf09d35c6b41bb48e9cfeb675003c62a8493a4b606b34"}