{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:QNE4SVEQJK4KCD6VR3MHY5O3BG","short_pith_number":"pith:QNE4SVEQ","canonical_record":{"source":{"id":"2605.22458","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-21T13:21:51Z","cross_cats_sorted":[],"title_canon_sha256":"ee4a6def181437ad8a284bf0b778a770a37c51051e57ca3aa6e0dab2e4acb7e4","abstract_canon_sha256":"ae9277d3a6ce2d0a668e42b26a408ab0c14a1b1c461c7b7bc7bc6cad4b7e685e"},"schema_version":"1.0"},"canonical_sha256":"8349c954904ab8a10fd58ed87c75db09ae2e0c6fbaaaf60059bbe4d9afc34380","source":{"kind":"arxiv","id":"2605.22458","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.22458","created_at":"2026-05-22T01:04:43Z"},{"alias_kind":"arxiv_version","alias_value":"2605.22458v1","created_at":"2026-05-22T01:04:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.22458","created_at":"2026-05-22T01:04:43Z"},{"alias_kind":"pith_short_12","alias_value":"QNE4SVEQJK4K","created_at":"2026-05-22T01:04:43Z"},{"alias_kind":"pith_short_16","alias_value":"QNE4SVEQJK4KCD6V","created_at":"2026-05-22T01:04:43Z"},{"alias_kind":"pith_short_8","alias_value":"QNE4SVEQ","created_at":"2026-05-22T01:04:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:QNE4SVEQJK4KCD6VR3MHY5O3BG","target":"record","payload":{"canonical_record":{"source":{"id":"2605.22458","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-21T13:21:51Z","cross_cats_sorted":[],"title_canon_sha256":"ee4a6def181437ad8a284bf0b778a770a37c51051e57ca3aa6e0dab2e4acb7e4","abstract_canon_sha256":"ae9277d3a6ce2d0a668e42b26a408ab0c14a1b1c461c7b7bc7bc6cad4b7e685e"},"schema_version":"1.0"},"canonical_sha256":"8349c954904ab8a10fd58ed87c75db09ae2e0c6fbaaaf60059bbe4d9afc34380","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:04:43.917822Z","signature_b64":"DUaZP9ayJKfeZSoAHRTw+GgmysZuhj5BdvjsXA+rC2Sw5Ehf4G3isk3VRY6Ohz7VYXOsyW/fr4TWP4hSFpNjCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8349c954904ab8a10fd58ed87c75db09ae2e0c6fbaaaf60059bbe4d9afc34380","last_reissued_at":"2026-05-22T01:04:43.917212Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:04:43.917212Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.22458","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-22T01:04:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WA/3+OpHXjjhawd5U6/gmXdHje+U/xYQpkZ6vxzOHU8OMIWUqiW8wPt4mob0L0TdXiIGw/FTc3Xt9haDJEnBDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T08:26:32.316475Z"},"content_sha256":"00a119147038d2f83f9f290e62898d93ab6ef90984bde9cf6a6c04391ba45ece","schema_version":"1.0","event_id":"sha256:00a119147038d2f83f9f290e62898d93ab6ef90984bde9cf6a6c04391ba45ece"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:QNE4SVEQJK4KCD6VR3MHY5O3BG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Complete Characterization of Heron Triangles with Two Perfect Square Sides and the All-Square Equivalence Condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Yangcheng Li","submitted_at":"2026-05-21T13:21:51Z","abstract_excerpt":"A Heron triangle is a triangle whose side lengths and area are all positive integers. If the greatest common divisor of the three side lengths is $1$, it is called a primitive Heron triangle. In this paper, we give an equivalent condition for Heron triangles with all three sides being perfect squares, which reduces to finding non-trivial rational points on a family of algebraic curves of genus $3$. This leads us to believe that only finitely many Heron triangles with three perfect square sides exist. Using a specific elliptic curve, we completely characterize all Heron triangles with two sides"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22458","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/2605.22458/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-22T01:04:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SLxFDxwMbrkV1HX0IL7ogWVy6tk1Fp5kaqLRauUEprN5SpR92UVMtmTQ8X6yec+ln6M5fp35Z+dLGTfLvrv/BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T08:26:32.317254Z"},"content_sha256":"e8f402e9e3752ab2adb07d0d536081bc04226481c5913dd9b497351bbe256919","schema_version":"1.0","event_id":"sha256:e8f402e9e3752ab2adb07d0d536081bc04226481c5913dd9b497351bbe256919"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QNE4SVEQJK4KCD6VR3MHY5O3BG/bundle.json","state_url":"https://pith.science/pith/QNE4SVEQJK4KCD6VR3MHY5O3BG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QNE4SVEQJK4KCD6VR3MHY5O3BG/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-10T08:26:32Z","links":{"resolver":"https://pith.science/pith/QNE4SVEQJK4KCD6VR3MHY5O3BG","bundle":"https://pith.science/pith/QNE4SVEQJK4KCD6VR3MHY5O3BG/bundle.json","state":"https://pith.science/pith/QNE4SVEQJK4KCD6VR3MHY5O3BG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QNE4SVEQJK4KCD6VR3MHY5O3BG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:QNE4SVEQJK4KCD6VR3MHY5O3BG","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":"ae9277d3a6ce2d0a668e42b26a408ab0c14a1b1c461c7b7bc7bc6cad4b7e685e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-21T13:21:51Z","title_canon_sha256":"ee4a6def181437ad8a284bf0b778a770a37c51051e57ca3aa6e0dab2e4acb7e4"},"schema_version":"1.0","source":{"id":"2605.22458","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.22458","created_at":"2026-05-22T01:04:43Z"},{"alias_kind":"arxiv_version","alias_value":"2605.22458v1","created_at":"2026-05-22T01:04:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.22458","created_at":"2026-05-22T01:04:43Z"},{"alias_kind":"pith_short_12","alias_value":"QNE4SVEQJK4K","created_at":"2026-05-22T01:04:43Z"},{"alias_kind":"pith_short_16","alias_value":"QNE4SVEQJK4KCD6V","created_at":"2026-05-22T01:04:43Z"},{"alias_kind":"pith_short_8","alias_value":"QNE4SVEQ","created_at":"2026-05-22T01:04:43Z"}],"graph_snapshots":[{"event_id":"sha256:e8f402e9e3752ab2adb07d0d536081bc04226481c5913dd9b497351bbe256919","target":"graph","created_at":"2026-05-22T01:04:43Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2605.22458/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A Heron triangle is a triangle whose side lengths and area are all positive integers. If the greatest common divisor of the three side lengths is $1$, it is called a primitive Heron triangle. In this paper, we give an equivalent condition for Heron triangles with all three sides being perfect squares, which reduces to finding non-trivial rational points on a family of algebraic curves of genus $3$. This leads us to believe that only finitely many Heron triangles with three perfect square sides exist. Using a specific elliptic curve, we completely characterize all Heron triangles with two sides","authors_text":"Yangcheng Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-21T13:21:51Z","title":"A Complete Characterization of Heron Triangles with Two Perfect Square Sides and the All-Square Equivalence Condition"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22458","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":"sha256:00a119147038d2f83f9f290e62898d93ab6ef90984bde9cf6a6c04391ba45ece","target":"record","created_at":"2026-05-22T01:04:43Z","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":"ae9277d3a6ce2d0a668e42b26a408ab0c14a1b1c461c7b7bc7bc6cad4b7e685e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-21T13:21:51Z","title_canon_sha256":"ee4a6def181437ad8a284bf0b778a770a37c51051e57ca3aa6e0dab2e4acb7e4"},"schema_version":"1.0","source":{"id":"2605.22458","kind":"arxiv","version":1}},"canonical_sha256":"8349c954904ab8a10fd58ed87c75db09ae2e0c6fbaaaf60059bbe4d9afc34380","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8349c954904ab8a10fd58ed87c75db09ae2e0c6fbaaaf60059bbe4d9afc34380","first_computed_at":"2026-05-22T01:04:43.917212Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T01:04:43.917212Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DUaZP9ayJKfeZSoAHRTw+GgmysZuhj5BdvjsXA+rC2Sw5Ehf4G3isk3VRY6Ohz7VYXOsyW/fr4TWP4hSFpNjCQ==","signature_status":"signed_v1","signed_at":"2026-05-22T01:04:43.917822Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.22458","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:00a119147038d2f83f9f290e62898d93ab6ef90984bde9cf6a6c04391ba45ece","sha256:e8f402e9e3752ab2adb07d0d536081bc04226481c5913dd9b497351bbe256919"],"state_sha256":"df72fd0d4a2fbc4b8e147efd4da2902ec2f88d7e7805883133ee245460575b44"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OumPk/8TwgvUqJFqsCAUgOFfDXkHhblJBZCIdy5gLNqcHi/1onwx6WPezTP7zhqGx7eiGukilqm2SAXLwmJPAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T08:26:32.322886Z","bundle_sha256":"4be763c2ae5d7d19dbadb029542610f267518fe2240aa943978bb376453910ba"}}