{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:ZF2SYHHVYQB7BIZCAHNTZ2HJIC","short_pith_number":"pith:ZF2SYHHV","canonical_record":{"source":{"id":"2606.07318","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-05T14:37:39Z","cross_cats_sorted":[],"title_canon_sha256":"e02deba39dc2551b9d8890ac1f3ea927c01c0f70edb028ed7c4cfda78df4e4f7","abstract_canon_sha256":"e538f7504032fc37a72b28ec118a01d99cb92e7923e4fb2af133f2cf4d41379e"},"schema_version":"1.0"},"canonical_sha256":"c9752c1cf5c403f0a32201db3ce8e940a0943740e6ef051b190a5ee5fe8a6e07","source":{"kind":"arxiv","id":"2606.07318","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.07318","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"arxiv_version","alias_value":"2606.07318v1","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.07318","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"pith_short_12","alias_value":"ZF2SYHHVYQB7","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"pith_short_16","alias_value":"ZF2SYHHVYQB7BIZC","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"pith_short_8","alias_value":"ZF2SYHHV","created_at":"2026-06-08T01:05:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:ZF2SYHHVYQB7BIZCAHNTZ2HJIC","target":"record","payload":{"canonical_record":{"source":{"id":"2606.07318","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-05T14:37:39Z","cross_cats_sorted":[],"title_canon_sha256":"e02deba39dc2551b9d8890ac1f3ea927c01c0f70edb028ed7c4cfda78df4e4f7","abstract_canon_sha256":"e538f7504032fc37a72b28ec118a01d99cb92e7923e4fb2af133f2cf4d41379e"},"schema_version":"1.0"},"canonical_sha256":"c9752c1cf5c403f0a32201db3ce8e940a0943740e6ef051b190a5ee5fe8a6e07","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:05:19.316152Z","signature_b64":"k2M9i5tLB74gQPP1jcdSQUzWPsuOu+cvJAR9UxFayMujVf7n8dPVdxOJ4Jn67ozLK1z/lO6g6FEfimGCAr+pBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9752c1cf5c403f0a32201db3ce8e940a0943740e6ef051b190a5ee5fe8a6e07","last_reissued_at":"2026-06-08T01:05:19.315370Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:05:19.315370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.07318","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":"2026-06-08T01:05:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nko/yeT3zC0irE7VqOjwgCbzC95j5YPLXFdoKbOD34ipsK0GQce9UK4F5wYx9GRMWAocOnE7WlKHxlxzZ/ukCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T12:35:21.918299Z"},"content_sha256":"0ad52b3c3e22403a31f3714333bc859a610be0f1c81b4068b6b9e0aa15024f40","schema_version":"1.0","event_id":"sha256:0ad52b3c3e22403a31f3714333bc859a610be0f1c81b4068b6b9e0aa15024f40"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:ZF2SYHHVYQB7BIZCAHNTZ2HJIC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the factorizations of integers via division algorithms for polynomials","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Adrian vasiu, Guram Donadze","submitted_at":"2026-06-05T14:37:39Z","abstract_excerpt":"We introduce and study several conditions related to the factorization problem of composite numbers. For this purpose, we employ cyclotomic polynomials, Sylvester resultants, and the Fermat equation. We show that the existence of a specific solution to the Fermat type equation in positive characteristic $p$ implies polynomial-time factorization of a composite natural number that is a multiple of $p$. We also show that such solutions do not exist for many semi-prime integers."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07318","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/2606.07318/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":"2026-06-08T01:05:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qv/VuRuCbfIjbo/bwWvV/ibEnZZmcYWKevdIH14+BN9vnvIqvdD4k9V2KSBs8PDAESvten2EVp6pSefY5pGDBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T12:35:21.918692Z"},"content_sha256":"5a73064993766ac8694eb44779ff9e910361ea8724c3a8c0dcea4df4a22dcedb","schema_version":"1.0","event_id":"sha256:5a73064993766ac8694eb44779ff9e910361ea8724c3a8c0dcea4df4a22dcedb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZF2SYHHVYQB7BIZCAHNTZ2HJIC/bundle.json","state_url":"https://pith.science/pith/ZF2SYHHVYQB7BIZCAHNTZ2HJIC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZF2SYHHVYQB7BIZCAHNTZ2HJIC/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":"2026-06-11T12:35:21Z","links":{"resolver":"https://pith.science/pith/ZF2SYHHVYQB7BIZCAHNTZ2HJIC","bundle":"https://pith.science/pith/ZF2SYHHVYQB7BIZCAHNTZ2HJIC/bundle.json","state":"https://pith.science/pith/ZF2SYHHVYQB7BIZCAHNTZ2HJIC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZF2SYHHVYQB7BIZCAHNTZ2HJIC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ZF2SYHHVYQB7BIZCAHNTZ2HJIC","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":"e538f7504032fc37a72b28ec118a01d99cb92e7923e4fb2af133f2cf4d41379e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-05T14:37:39Z","title_canon_sha256":"e02deba39dc2551b9d8890ac1f3ea927c01c0f70edb028ed7c4cfda78df4e4f7"},"schema_version":"1.0","source":{"id":"2606.07318","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.07318","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"arxiv_version","alias_value":"2606.07318v1","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.07318","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"pith_short_12","alias_value":"ZF2SYHHVYQB7","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"pith_short_16","alias_value":"ZF2SYHHVYQB7BIZC","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"pith_short_8","alias_value":"ZF2SYHHV","created_at":"2026-06-08T01:05:19Z"}],"graph_snapshots":[{"event_id":"sha256:5a73064993766ac8694eb44779ff9e910361ea8724c3a8c0dcea4df4a22dcedb","target":"graph","created_at":"2026-06-08T01:05:19Z","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/2606.07318/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce and study several conditions related to the factorization problem of composite numbers. For this purpose, we employ cyclotomic polynomials, Sylvester resultants, and the Fermat equation. We show that the existence of a specific solution to the Fermat type equation in positive characteristic $p$ implies polynomial-time factorization of a composite natural number that is a multiple of $p$. We also show that such solutions do not exist for many semi-prime integers.","authors_text":"Adrian vasiu, Guram Donadze","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-05T14:37:39Z","title":"On the factorizations of integers via division algorithms for polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07318","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":"sha256:0ad52b3c3e22403a31f3714333bc859a610be0f1c81b4068b6b9e0aa15024f40","target":"record","created_at":"2026-06-08T01:05:19Z","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":"e538f7504032fc37a72b28ec118a01d99cb92e7923e4fb2af133f2cf4d41379e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-05T14:37:39Z","title_canon_sha256":"e02deba39dc2551b9d8890ac1f3ea927c01c0f70edb028ed7c4cfda78df4e4f7"},"schema_version":"1.0","source":{"id":"2606.07318","kind":"arxiv","version":1}},"canonical_sha256":"c9752c1cf5c403f0a32201db3ce8e940a0943740e6ef051b190a5ee5fe8a6e07","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c9752c1cf5c403f0a32201db3ce8e940a0943740e6ef051b190a5ee5fe8a6e07","first_computed_at":"2026-06-08T01:05:19.315370Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-08T01:05:19.315370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"k2M9i5tLB74gQPP1jcdSQUzWPsuOu+cvJAR9UxFayMujVf7n8dPVdxOJ4Jn67ozLK1z/lO6g6FEfimGCAr+pBw==","signature_status":"signed_v1","signed_at":"2026-06-08T01:05:19.316152Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.07318","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0ad52b3c3e22403a31f3714333bc859a610be0f1c81b4068b6b9e0aa15024f40","sha256:5a73064993766ac8694eb44779ff9e910361ea8724c3a8c0dcea4df4a22dcedb"],"state_sha256":"c4bf156c0e19f93d75db25976ad89c53f4612828cd21f6fd66ebbf196f2ce3a1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2GVRziANSbfiDjtxETFMzGY9Tf9lPXIqN+cEw99mRXQGmbFJHz+Y4BkaEOu+Nh5YwybdE/xoI/dVu6Mh6YB0DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T12:35:21.920714Z","bundle_sha256":"5ccb4284c4435474fc124f853c1fc7b871051fc3d969853db15dffa18ddb13d1"}}