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One can compare this to the similar problem of counting solutions to $F(\\mathbf{x})=0$ without the congruence condition. It turns out that adding the congruence condition sometimes gives a very different main term than the homogeneous case. In particular, there are examples where the number of primitive solutions to the problem is $0$, while the number of unrestricted solutions is nonzero."},"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":"1704.00502","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-04-03T09:52:17Z","cross_cats_sorted":[],"title_canon_sha256":"f929fa0e82d49c992976787de08134efb98e073c50ce11b5ed7039cf0d0208fb","abstract_canon_sha256":"2a336ffc68d6f8d5f15b24009a81b719aa4271e6a01c8469250835a30ed1d208"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:22.168872Z","signature_b64":"aCcxck8mzk3RAif+dZuvD0Q8+2DYyWDtEPZqbAFKOKM+9bFLgsZMueG8uG4FBFuI61gz5GUjJLKVva/x3bIlCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f1bbaa0e2323fda4ab2f2191765f6b57caecd042a5bc314496e895ee403ff09","last_reissued_at":"2026-05-18T00:47:22.168361Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:22.168361Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weak approximation results for quadratic forms in four variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Sofia Lindqvist","submitted_at":"2017-04-03T09:52:17Z","abstract_excerpt":"Let $F$ be a quadratic form in four variables, let $m\\in\\mathbb{N}$ and let $\\mathbf{k}\\in \\mathbb{Z}^4$. We count integer solutions to $F(\\mathbf{x})=0$ with $\\mathbf{x}\\equiv \\mathbf{k}\\:\\mathrm{mod}(m)$. One can compare this to the similar problem of counting solutions to $F(\\mathbf{x})=0$ without the congruence condition. It turns out that adding the congruence condition sometimes gives a very different main term than the homogeneous case. 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