{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:C57RVIMI2CKWAJKQCWMCWVD33G","short_pith_number":"pith:C57RVIMI","canonical_record":{"source":{"id":"1711.05673","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-11-15T17:07:40Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"67861ebb3781fcd1ba1c35b3d0a7f9b2c8b4a5373059364dd2df52ff3b89684a","abstract_canon_sha256":"2a4c1f057a3791fd00e987e71eab59f18716b1499cbb0e2befd95ba978163dc1"},"schema_version":"1.0"},"canonical_sha256":"177f1aa188d09560255015982b547bd993b83b9f6920618ecbde731aa7d71d43","source":{"kind":"arxiv","id":"1711.05673","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.05673","created_at":"2026-05-18T00:30:29Z"},{"alias_kind":"arxiv_version","alias_value":"1711.05673v1","created_at":"2026-05-18T00:30:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.05673","created_at":"2026-05-18T00:30:29Z"},{"alias_kind":"pith_short_12","alias_value":"C57RVIMI2CKW","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"C57RVIMI2CKWAJKQ","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"C57RVIMI","created_at":"2026-05-18T12:31:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:C57RVIMI2CKWAJKQCWMCWVD33G","target":"record","payload":{"canonical_record":{"source":{"id":"1711.05673","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-11-15T17:07:40Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"67861ebb3781fcd1ba1c35b3d0a7f9b2c8b4a5373059364dd2df52ff3b89684a","abstract_canon_sha256":"2a4c1f057a3791fd00e987e71eab59f18716b1499cbb0e2befd95ba978163dc1"},"schema_version":"1.0"},"canonical_sha256":"177f1aa188d09560255015982b547bd993b83b9f6920618ecbde731aa7d71d43","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:29.246519Z","signature_b64":"eu3NGC+jCjOMONYXs04E0LxjsSNk+PI7vhcjIdYQBM0LW8Ru0HrsRdSvlfVsHl8kVPVarQ4kWksLuE0uCRMhCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"177f1aa188d09560255015982b547bd993b83b9f6920618ecbde731aa7d71d43","last_reissued_at":"2026-05-18T00:30:29.245924Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:29.245924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1711.05673","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":"2026-05-18T00:30:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7SdZCS2HGdoz1SxOaC6kozUCKT4xkpl+wpZwF9yHE7Nk2o1YH0tZRWbOKPaDDAQpDrSilsZrA9gzSmHgYIijDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T06:36:42.936368Z"},"content_sha256":"c86ceb4b6b88a21bd2dc8da7b0993486ffd50fb11f027e3edeb2d7106082cff6","schema_version":"1.0","event_id":"sha256:c86ceb4b6b88a21bd2dc8da7b0993486ffd50fb11f027e3edeb2d7106082cff6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:C57RVIMI2CKWAJKQCWMCWVD33G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting factorisations of monomials over rings of integers modulo $N$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"James Wright, Jonathan Hickman","submitted_at":"2017-11-15T17:07:40Z","abstract_excerpt":"A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\\mathbb{Z}/p^{\\alpha}\\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number of solutions to a certain system of polynomial congruences. The method also applies to more general systems of polynomial congruences that satisfy a non-degeneracy hypothesis."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.05673","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":""},"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":"2026-05-18T00:30:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XgzPAZk7xmPho2zXXfk2qS8RTeS9scQOqkNz+oxV+tYidrJiwPWcJktHYnDZNd/hx77xPCE09UE57IrlirfxCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T06:36:42.937005Z"},"content_sha256":"20214da99a29288cf1849db4d0a3454bc39636d892a739d73e1f2a574b2dc830","schema_version":"1.0","event_id":"sha256:20214da99a29288cf1849db4d0a3454bc39636d892a739d73e1f2a574b2dc830"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/C57RVIMI2CKWAJKQCWMCWVD33G/bundle.json","state_url":"https://pith.science/pith/C57RVIMI2CKWAJKQCWMCWVD33G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/C57RVIMI2CKWAJKQCWMCWVD33G/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":"2026-05-25T06:36:42Z","links":{"resolver":"https://pith.science/pith/C57RVIMI2CKWAJKQCWMCWVD33G","bundle":"https://pith.science/pith/C57RVIMI2CKWAJKQCWMCWVD33G/bundle.json","state":"https://pith.science/pith/C57RVIMI2CKWAJKQCWMCWVD33G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/C57RVIMI2CKWAJKQCWMCWVD33G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:C57RVIMI2CKWAJKQCWMCWVD33G","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":"2a4c1f057a3791fd00e987e71eab59f18716b1499cbb0e2befd95ba978163dc1","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-11-15T17:07:40Z","title_canon_sha256":"67861ebb3781fcd1ba1c35b3d0a7f9b2c8b4a5373059364dd2df52ff3b89684a"},"schema_version":"1.0","source":{"id":"1711.05673","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.05673","created_at":"2026-05-18T00:30:29Z"},{"alias_kind":"arxiv_version","alias_value":"1711.05673v1","created_at":"2026-05-18T00:30:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.05673","created_at":"2026-05-18T00:30:29Z"},{"alias_kind":"pith_short_12","alias_value":"C57RVIMI2CKW","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"C57RVIMI2CKWAJKQ","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"C57RVIMI","created_at":"2026-05-18T12:31:08Z"}],"graph_snapshots":[{"event_id":"sha256:20214da99a29288cf1849db4d0a3454bc39636d892a739d73e1f2a574b2dc830","target":"graph","created_at":"2026-05-18T00:30:29Z","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":"A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\\mathbb{Z}/p^{\\alpha}\\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number of solutions to a certain system of polynomial congruences. The method also applies to more general systems of polynomial congruences that satisfy a non-degeneracy hypothesis.","authors_text":"James Wright, Jonathan Hickman","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-11-15T17:07:40Z","title":"Counting factorisations of monomials over rings of integers modulo $N$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.05673","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:c86ceb4b6b88a21bd2dc8da7b0993486ffd50fb11f027e3edeb2d7106082cff6","target":"record","created_at":"2026-05-18T00:30:29Z","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":"2a4c1f057a3791fd00e987e71eab59f18716b1499cbb0e2befd95ba978163dc1","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-11-15T17:07:40Z","title_canon_sha256":"67861ebb3781fcd1ba1c35b3d0a7f9b2c8b4a5373059364dd2df52ff3b89684a"},"schema_version":"1.0","source":{"id":"1711.05673","kind":"arxiv","version":1}},"canonical_sha256":"177f1aa188d09560255015982b547bd993b83b9f6920618ecbde731aa7d71d43","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"177f1aa188d09560255015982b547bd993b83b9f6920618ecbde731aa7d71d43","first_computed_at":"2026-05-18T00:30:29.245924Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:30:29.245924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eu3NGC+jCjOMONYXs04E0LxjsSNk+PI7vhcjIdYQBM0LW8Ru0HrsRdSvlfVsHl8kVPVarQ4kWksLuE0uCRMhCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:30:29.246519Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.05673","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c86ceb4b6b88a21bd2dc8da7b0993486ffd50fb11f027e3edeb2d7106082cff6","sha256:20214da99a29288cf1849db4d0a3454bc39636d892a739d73e1f2a574b2dc830"],"state_sha256":"ff009aac3802bac721d4b545872c14f4bd5481fb723c1eaba852361a9e9d337d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gwbkVyPLdW6mYDdxoOhIXUe5+26OmYqyolvyT1axCa52nyeGG3eEL6bLijLJ04BMKd13OqgkgPtxuPzeCyUcBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T06:36:42.940754Z","bundle_sha256":"b9e89ec2e9055c1a8b22954057142ef9cfb711dce5b49b2141f2213c8fbcd069"}}