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Moreover, let $\\mathcal{S}_R$ be th"},"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":"1603.06030","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-03-19T01:57:02Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"a98c88626b97aca67b05b58dfbdae7f153f0be757ed9d9b87ffe6f190b82da0b","abstract_canon_sha256":"a184020a36d54b7703ba11f9c19aae4f8cf2d1fc20940e9a0afb046881dd5e47"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:51.948152Z","signature_b64":"rHyku59Izm85siuEU7hQ667CGsKvAlukp4kRasfXR3j/5aGLwWXJ2TbugW63lL1kRwYhat9fr3WsRX0jEhG6Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"657ff3cb3a0f89813925229cce8a15b0651f0508084652459cdaf05c99e9e904","last_reissued_at":"2026-05-18T01:18:51.947666Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:51.947666Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Davenport constant of the multiplicative semigroup of the ring $\\mathbb{Z}_{n_1}\\oplus\\cdots\\oplus \\mathbb{Z}_{n_r}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Guoqing Wang, Weidong Gao","submitted_at":"2016-03-19T01:57:02Z","abstract_excerpt":"Given a finite commutative semigroup $\\mathcal{S}$ (written additively), denoted by ${\\rm D}(\\mathcal{S})$ the Davenport constant of $\\mathcal{S}$, namely the least positive integer $\\ell$ such that for any $\\ell$ elements $s_1,\\ldots,s_{\\ell}\\in \\mathcal{S}$ there exists a set $I\\subsetneq [1,\\ell]$ for which $\\sum_{i\\in I} s_i=\\sum_{i=1}^{\\ell} s_i$.\n  Then, for any integers $r\\geq 1, n_1,\\ldots,n_r>1$, let $R=\\mathbb{Z}_{n_1}\\oplus\\cdots\\oplus \\mathbb{Z}_{n_r}$ be the direct sum of these $r$ residue class rings $\\mathbb{Z}_{n_1}, \\ldots,\\mathbb{Z}_{n_r}$. 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