{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:KJFLUW62BAQ27JFX5XDUGGD6R4","short_pith_number":"pith:KJFLUW62","schema_version":"1.0","canonical_sha256":"524aba5bda0821afa4b7edc743187e8f0799b05bc42c2bf8af6fade9ddb50e79","source":{"kind":"arxiv","id":"1403.0542","version":2},"attestation_state":"computed","paper":{"title":"Counting solutions without zeros or repetitions of a linear congruence and rarefaction in b-multiplicative sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexandre Aksenov","submitted_at":"2014-03-03T19:46:59Z","abstract_excerpt":"Consider a strongly $b$-multiplicative sequence and a prime $p$. Studying its $p$-rarefaction consists in characterizing the asymptotic behaviour of the sums of the first terms indexed by the multiples of $p$. The integer values of the \"norm\" $3$-variate polynomial $\\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2)\\!:=\\!\\prod_{j=1}^{p-1}\\left(Y_0{+}\\zeta_p^{i_1j}Y_1{+}\\zeta_p^{i_2j}Y_2\\right),$ where $\\zeta_p$ is a primitive $p$-th root of unity, and $i_1,i_2{\\in}\\{1,2,\\dots,p{-}1\\},$ determine this asymptotic behaviour. It will be shown that a combinatorial method can be applied to $\\mathcal N_{p,i_1,i_2}("},"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":"1403.0542","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-03-03T19:46:59Z","cross_cats_sorted":[],"title_canon_sha256":"4b2bc9e114cad53676f43f075a335d54ca8bc1b2418cc33cbd0e05c8791242bd","abstract_canon_sha256":"2f1c41721a9d42e0c724583cc2b099353082fea5d38c90912521f3072a9d305f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:05.515468Z","signature_b64":"58+6E0vp/5e7x/sDAG7r9A/MBgSnx4v5xjGRxRemz4zWUYrSC4pfw03SXq1TA/yh9gjcpBDeacOTuJpV6zb4Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"524aba5bda0821afa4b7edc743187e8f0799b05bc42c2bf8af6fade9ddb50e79","last_reissued_at":"2026-05-18T01:21:05.514912Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:05.514912Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counting solutions without zeros or repetitions of a linear congruence and rarefaction in b-multiplicative sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexandre Aksenov","submitted_at":"2014-03-03T19:46:59Z","abstract_excerpt":"Consider a strongly $b$-multiplicative sequence and a prime $p$. Studying its $p$-rarefaction consists in characterizing the asymptotic behaviour of the sums of the first terms indexed by the multiples of $p$. The integer values of the \"norm\" $3$-variate polynomial $\\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2)\\!:=\\!\\prod_{j=1}^{p-1}\\left(Y_0{+}\\zeta_p^{i_1j}Y_1{+}\\zeta_p^{i_2j}Y_2\\right),$ where $\\zeta_p$ is a primitive $p$-th root of unity, and $i_1,i_2{\\in}\\{1,2,\\dots,p{-}1\\},$ determine this asymptotic behaviour. It will be shown that a combinatorial method can be applied to $\\mathcal N_{p,i_1,i_2}("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.0542","kind":"arxiv","version":2},"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":"1403.0542","created_at":"2026-05-18T01:21:05.514995+00:00"},{"alias_kind":"arxiv_version","alias_value":"1403.0542v2","created_at":"2026-05-18T01:21:05.514995+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.0542","created_at":"2026-05-18T01:21:05.514995+00:00"},{"alias_kind":"pith_short_12","alias_value":"KJFLUW62BAQ2","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_16","alias_value":"KJFLUW62BAQ27JFX","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_8","alias_value":"KJFLUW62","created_at":"2026-05-18T12:28:35.611951+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/KJFLUW62BAQ27JFX5XDUGGD6R4","json":"https://pith.science/pith/KJFLUW62BAQ27JFX5XDUGGD6R4.json","graph_json":"https://pith.science/api/pith-number/KJFLUW62BAQ27JFX5XDUGGD6R4/graph.json","events_json":"https://pith.science/api/pith-number/KJFLUW62BAQ27JFX5XDUGGD6R4/events.json","paper":"https://pith.science/paper/KJFLUW62"},"agent_actions":{"view_html":"https://pith.science/pith/KJFLUW62BAQ27JFX5XDUGGD6R4","download_json":"https://pith.science/pith/KJFLUW62BAQ27JFX5XDUGGD6R4.json","view_paper":"https://pith.science/paper/KJFLUW62","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1403.0542&json=true","fetch_graph":"https://pith.science/api/pith-number/KJFLUW62BAQ27JFX5XDUGGD6R4/graph.json","fetch_events":"https://pith.science/api/pith-number/KJFLUW62BAQ27JFX5XDUGGD6R4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KJFLUW62BAQ27JFX5XDUGGD6R4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KJFLUW62BAQ27JFX5XDUGGD6R4/action/storage_attestation","attest_author":"https://pith.science/pith/KJFLUW62BAQ27JFX5XDUGGD6R4/action/author_attestation","sign_citation":"https://pith.science/pith/KJFLUW62BAQ27JFX5XDUGGD6R4/action/citation_signature","submit_replication":"https://pith.science/pith/KJFLUW62BAQ27JFX5XDUGGD6R4/action/replication_record"}},"created_at":"2026-05-18T01:21:05.514995+00:00","updated_at":"2026-05-18T01:21:05.514995+00:00"}