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Let $\\mathcal{Q}_N(d_K)$ be the set of primitive positive definite binary quadratic forms of discriminant $d_K$ whose leading coefficients are relatively prime to $N$. We adopt an equivalence relation $\\sim_\\mathfrak{n}$ on $\\mathcal{Q}_N(d_K)$ so that the set of equivalence classes $\\mathcal{Q}_N(d_K)/\\sim_\\mathfrak{n}$ can be regarded as a group isomorphic to the ray class group of $K$ modulo $\\mathfrak{n}$. We further pres"},"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":"1905.11690","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-05-28T09:02:07Z","cross_cats_sorted":[],"title_canon_sha256":"ee0f301455836a9f3ef8440cc72d92c454d562fb23068361d4e9f9dce42ab515","abstract_canon_sha256":"8e7071cb25facb4e72d647f5f1a97b0588b4ab2c8a8447de9c1d5fb3f43aefcb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:52.643134Z","signature_b64":"IRUZWtAAt+IfkoZ77d8u+Ci8rsbIApVvwil8q/otQQhQLZpTcdg87rgkE6scuLTRt0tylp6HVpj0/PP3zN58Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"800efd0240b7de1e1345db223c77a46707a15f46a96c51e9e69dc8d1e9ec49ce","last_reissued_at":"2026-05-17T23:44:52.642502Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:52.642502Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On some extension of Gauss' work and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dong Hwa Shin, Ho Yun Jung, Ja Kyung Koo","submitted_at":"2019-05-28T09:02:07Z","abstract_excerpt":"Let $K$ be an imaginary quadratic field of discriminant $d_K$, and let $\\mathfrak{n}$ be a nontrivial integral ideal of $K$ in which $N$ is the smallest positive integer. Let $\\mathcal{Q}_N(d_K)$ be the set of primitive positive definite binary quadratic forms of discriminant $d_K$ whose leading coefficients are relatively prime to $N$. We adopt an equivalence relation $\\sim_\\mathfrak{n}$ on $\\mathcal{Q}_N(d_K)$ so that the set of equivalence classes $\\mathcal{Q}_N(d_K)/\\sim_\\mathfrak{n}$ can be regarded as a group isomorphic to the ray class group of $K$ modulo $\\mathfrak{n}$. 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