{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:IQSNSCFM5WQJSRMPFMCUWGZRGW","short_pith_number":"pith:IQSNSCFM","schema_version":"1.0","canonical_sha256":"4424d908aceda099458f2b054b1b31358aeea3f7e9c9a24a4b3959ca89cc3df5","source":{"kind":"arxiv","id":"1811.03914","version":1},"attestation_state":"computed","paper":{"title":"The subsums of zero-sum free sequences in finite cyclic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"S\\'avio Ribas","submitted_at":"2018-11-09T14:17:51Z","abstract_excerpt":"Let $\\mathbb Z_n$ be the cyclic group of order $n \\ge 3$ additively written. S. Savchev \\& F. Chen (2007) proved that for each zero-sum free sequence $S = a_1 \\bullet \\dots \\bullet a_t$ over $\\mathbb Z_n$ of length $t > n/2$, there is an integer $g$ coprime to $n$ such that, if $\\overline{r}$ denotes the least positive integer in the congruence class $r$ modulo $n$, then $\\sum_{i=1}^t \\overline{ga_i} < n$. Under the same hypothesis, in this paper we show that $$\\left\\{ \\sum_{i \\in \\Lambda} \\overline{ga_i} \\;\\; \\Bigg| \\;\\; \\Lambda \\subset \\{1,2,\\dots,t\\} \\right\\} = \\left\\{ 1, 2, \\dots, \\sum_{i="},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1811.03914","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-11-09T14:17:51Z","cross_cats_sorted":[],"title_canon_sha256":"c465b5e374b22bc5cc1fd46b21a4c5b7d85a8cee6d3a4e9a9ec733b4f27713a8","abstract_canon_sha256":"18d316e27d647c068ae66ec563249923b6763341e7b9bf34b718541310c1ad7d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:11.592946Z","signature_b64":"OABE7nnGdLxZS3RE3FSrZ8PIYTu5rjnAwkrUf0GcGf+RvMU/No7czeThD6gc3+U9uJp6p+AhSovWttQZ5RDtCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4424d908aceda099458f2b054b1b31358aeea3f7e9c9a24a4b3959ca89cc3df5","last_reissued_at":"2026-05-18T00:01:11.592407Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:11.592407Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The subsums of zero-sum free sequences in finite cyclic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"S\\'avio Ribas","submitted_at":"2018-11-09T14:17:51Z","abstract_excerpt":"Let $\\mathbb Z_n$ be the cyclic group of order $n \\ge 3$ additively written. S. Savchev \\& F. Chen (2007) proved that for each zero-sum free sequence $S = a_1 \\bullet \\dots \\bullet a_t$ over $\\mathbb Z_n$ of length $t > n/2$, there is an integer $g$ coprime to $n$ such that, if $\\overline{r}$ denotes the least positive integer in the congruence class $r$ modulo $n$, then $\\sum_{i=1}^t \\overline{ga_i} < n$. Under the same hypothesis, in this paper we show that $$\\left\\{ \\sum_{i \\in \\Lambda} \\overline{ga_i} \\;\\; \\Bigg| \\;\\; \\Lambda \\subset \\{1,2,\\dots,t\\} \\right\\} = \\left\\{ 1, 2, \\dots, \\sum_{i="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.03914","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1811.03914","created_at":"2026-05-18T00:01:11.592488+00:00"},{"alias_kind":"arxiv_version","alias_value":"1811.03914v1","created_at":"2026-05-18T00:01:11.592488+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.03914","created_at":"2026-05-18T00:01:11.592488+00:00"},{"alias_kind":"pith_short_12","alias_value":"IQSNSCFM5WQJ","created_at":"2026-05-18T12:32:31.084164+00:00"},{"alias_kind":"pith_short_16","alias_value":"IQSNSCFM5WQJSRMP","created_at":"2026-05-18T12:32:31.084164+00:00"},{"alias_kind":"pith_short_8","alias_value":"IQSNSCFM","created_at":"2026-05-18T12:32:31.084164+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IQSNSCFM5WQJSRMPFMCUWGZRGW","json":"https://pith.science/pith/IQSNSCFM5WQJSRMPFMCUWGZRGW.json","graph_json":"https://pith.science/api/pith-number/IQSNSCFM5WQJSRMPFMCUWGZRGW/graph.json","events_json":"https://pith.science/api/pith-number/IQSNSCFM5WQJSRMPFMCUWGZRGW/events.json","paper":"https://pith.science/paper/IQSNSCFM"},"agent_actions":{"view_html":"https://pith.science/pith/IQSNSCFM5WQJSRMPFMCUWGZRGW","download_json":"https://pith.science/pith/IQSNSCFM5WQJSRMPFMCUWGZRGW.json","view_paper":"https://pith.science/paper/IQSNSCFM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1811.03914&json=true","fetch_graph":"https://pith.science/api/pith-number/IQSNSCFM5WQJSRMPFMCUWGZRGW/graph.json","fetch_events":"https://pith.science/api/pith-number/IQSNSCFM5WQJSRMPFMCUWGZRGW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IQSNSCFM5WQJSRMPFMCUWGZRGW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IQSNSCFM5WQJSRMPFMCUWGZRGW/action/storage_attestation","attest_author":"https://pith.science/pith/IQSNSCFM5WQJSRMPFMCUWGZRGW/action/author_attestation","sign_citation":"https://pith.science/pith/IQSNSCFM5WQJSRMPFMCUWGZRGW/action/citation_signature","submit_replication":"https://pith.science/pith/IQSNSCFM5WQJSRMPFMCUWGZRGW/action/replication_record"}},"created_at":"2026-05-18T00:01:11.592488+00:00","updated_at":"2026-05-18T00:01:11.592488+00:00"}