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As a corollary of our first main result, we verify the conjecture for abelian groups of the form $C_{p^m}\\oplus C_p, C_{p^m}\\oplus C_q, C_{p^m}\\oplus C_q^2$, $C_{p^m}\\oplus C_r^n$ where $p,q$ are distinct primes and $r\\in\\{2,3\\}$. In our second main result we verify that $K_1(G) = K_1^*(G)$ for groups of the form $C_r\\oplus C_{p^m}\\oplus C_p, C_{rp^mq}$ and $C_r\\oplus C_p "},"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":"1301.1401","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-01-08T03:01:13Z","cross_cats_sorted":[],"title_canon_sha256":"341f1c3d920710f67eb153613437b9539374988199b8007548ce8415ac177d0e","abstract_canon_sha256":"b930c5749abe087cdd5f709e02e56cd21a8924b0b4c46a0cd5f67c206656790d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:36:57.602669Z","signature_b64":"wb6sTAne5XgiVOqEom4ch8nvY0wG8fEMOD7Zx5QPj/p5Tgg9l2V6sqC2GPxN4FsO96dxvYvQRnKEmtNT0eMnAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b303e41b3f5f4d7fb792e4f7356f57edafedb87d76e62d5ae18da99a917533f5","last_reissued_at":"2026-05-18T03:36:57.602218Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:36:57.602218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the maximal cross number of unique factorization indexed multisets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Daniel Kriz","submitted_at":"2013-01-08T03:01:13Z","abstract_excerpt":"In this paper, we study a conjecture of Gao and Wang concerning a proposed formula $K_1^*(G)$ for the maximal cross number $K_1(G)$ taken over all unique factorization indexed multisets over a given finite abelian group $G$. As a corollary of our first main result, we verify the conjecture for abelian groups of the form $C_{p^m}\\oplus C_p, C_{p^m}\\oplus C_q, C_{p^m}\\oplus C_q^2$, $C_{p^m}\\oplus C_r^n$ where $p,q$ are distinct primes and $r\\in\\{2,3\\}$. 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