{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:62JTCQ5QKATDVMQRXWVII6BK5C","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":"e7c987995581dc5233fd2f0c1cfa7ddb24d6b93fbef5ae5ff207b93127b3e028","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-21T15:28:58Z","title_canon_sha256":"c920b9bcae94e6a77aaa59a379cb6b2037fbc440087a0938c1c26d95c730767e"},"schema_version":"1.0","source":{"id":"1509.06272","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.06272","created_at":"2026-05-18T01:32:35Z"},{"alias_kind":"arxiv_version","alias_value":"1509.06272v1","created_at":"2026-05-18T01:32:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.06272","created_at":"2026-05-18T01:32:35Z"},{"alias_kind":"pith_short_12","alias_value":"62JTCQ5QKATD","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"62JTCQ5QKATDVMQR","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"62JTCQ5Q","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:8f7cace6dc80ca3c6952d71caa604c254921dfdf237fa64ace1b37ce58f3ba0d","target":"graph","created_at":"2026-05-18T01:32:35Z","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":"Let $R$ be a commutative ring with $1\\in R$ and $R^{\\ast}$ be the multiplicative group of its units. In 1969, Nagell introduced the exceptional unit $u$ if both $u$ and $1-u$ belong to $R^{\\ast}$. Let $\\mathbb{Z}_n$ be the ring of residue classes modulo $n$. In this paper, given an integer $k\\ge 2$, we obtain an exact formula for the number of ways to represent each element of $ \\mathbb{Z}_n$ as the sum of $k$ exceptional units. This generalizes a recent result of J. W. Sander for the case $k=2$.","authors_text":"Qing-Qing Zhao, Quan-Hui Yang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-21T15:28:58Z","title":"On the sumsets of exceptional units in $\\mathbb{Z}_n$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.06272","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":"sha256:4363f7d4f9ea0b1c797020c774b7ac19a2dcb72a1e0dc1b5acaf312269503bf1","target":"record","created_at":"2026-05-18T01:32:35Z","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":"e7c987995581dc5233fd2f0c1cfa7ddb24d6b93fbef5ae5ff207b93127b3e028","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-21T15:28:58Z","title_canon_sha256":"c920b9bcae94e6a77aaa59a379cb6b2037fbc440087a0938c1c26d95c730767e"},"schema_version":"1.0","source":{"id":"1509.06272","kind":"arxiv","version":1}},"canonical_sha256":"f6933143b050263ab211bdaa84782ae8bd9ce75838128727884aed56139ec815","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f6933143b050263ab211bdaa84782ae8bd9ce75838128727884aed56139ec815","first_computed_at":"2026-05-18T01:32:35.441817Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:35.441817Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9kQbnE66ar6EWW0af1ku3f9aUwbo8ij3rRhfnlops08pQ2qRjq4yE7AD5mHjJvtBof7CeFpIpajJ2IC4Uwx8Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:35.442350Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.06272","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4363f7d4f9ea0b1c797020c774b7ac19a2dcb72a1e0dc1b5acaf312269503bf1","sha256:8f7cace6dc80ca3c6952d71caa604c254921dfdf237fa64ace1b37ce58f3ba0d"],"state_sha256":"cfbd26da00a7058d4c19deb2999d62ec976bef96944064d56e2a397e84a39ce7"}