{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:GA2CEM5IIV6ZEEZU4ZYSJIYARH","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":"6d4a43b467f8e256e73fe0a90af59847100ab66473ab845f929c7acab28603e6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-20T11:31:09Z","title_canon_sha256":"076f4637f7566f5bb0aa3645db6bc92d40707a07a87fcb808e77a25fc627a999"},"schema_version":"1.0","source":{"id":"1603.06205","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.06205","created_at":"2026-05-18T01:18:50Z"},{"alias_kind":"arxiv_version","alias_value":"1603.06205v1","created_at":"2026-05-18T01:18:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.06205","created_at":"2026-05-18T01:18:50Z"},{"alias_kind":"pith_short_12","alias_value":"GA2CEM5IIV6Z","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"GA2CEM5IIV6ZEEZU","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"GA2CEM5I","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:e092776c35520b306fdb0f40a0e5924444b22ec5aa928987b5c52f19de9fe7af","target":"graph","created_at":"2026-05-18T01:18:50Z","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 paper we examine the diophantine equation $x^k-y^k=x-y$ where $k$ is a positive integer $\\geq 2$, and consider its applications. While the complete solution of the equation $x^k-y^k=x-y$ in positive rational numbers is already known when $k=2$ or $3$, till now only one numerical solution of the equation in positive rational numbers has been published when $k=4$, and no nontrivial solution is known when $k \\geq 5$. We describe a method of generating infinitely many positive rational solutions of the equation when $k=4$. We use the positive rational solutions of the equation with $k=2,\\,","authors_text":"Ajai Choudhry, Jaros{\\l}aw Wr\\'oblewski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-20T11:31:09Z","title":"An Ancient Diophantine Equation with applications to Numerical Curios and Geometric Series"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06205","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:e2d58ddf5f186c53548a263668bd229cf08e76825709dae3d5b62e1a20be293f","target":"record","created_at":"2026-05-18T01:18:50Z","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":"6d4a43b467f8e256e73fe0a90af59847100ab66473ab845f929c7acab28603e6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-20T11:31:09Z","title_canon_sha256":"076f4637f7566f5bb0aa3645db6bc92d40707a07a87fcb808e77a25fc627a999"},"schema_version":"1.0","source":{"id":"1603.06205","kind":"arxiv","version":1}},"canonical_sha256":"30342233a8457d921334e67124a30089efb29aac8a52616d994b1bc2ceb31119","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"30342233a8457d921334e67124a30089efb29aac8a52616d994b1bc2ceb31119","first_computed_at":"2026-05-18T01:18:50.032091Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:18:50.032091Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6c6TdpgmhDsL5wW/KuNeyLE5kWiQgJI/rqcloz6+ILSvlrZCNoxenn2vrlAOvxd691prHhEZ1X+GA75JQmAsCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:18:50.032531Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.06205","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e2d58ddf5f186c53548a263668bd229cf08e76825709dae3d5b62e1a20be293f","sha256:e092776c35520b306fdb0f40a0e5924444b22ec5aa928987b5c52f19de9fe7af"],"state_sha256":"925af4849ff5bbc943f61925e4429eefff94c86c39f31970d5805d16cb57f87f"}