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For $a,b,c,d,n\\in\\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$ $(x,y,z,w\\in\\Bbb Z$). In this paper we reveal some connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$."},"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":"1511.06177","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-11-19T14:15:42Z","cross_cats_sorted":[],"title_canon_sha256":"5eae5deb0e5ad00cf44a278f50187e172ca224d6518c93b937ca671f4333ce62","abstract_canon_sha256":"5782a6965b36828d8754a1fc72c590c7782ed85a2ea6b552cc7bc31f0459b2a4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:11.062237Z","signature_b64":"TSvh0n8dUV1q/uHjI55Pn843Py5cnN8UNYmuxRJMxrhfPU6QHyo4n9zwwYcmMxaoAu1Ptfxw/TAwsnx3Pnf9Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"56d2b204de66f1fd2e92f12e7e3dfa23d0134174291916386b2dc3b16c528c56","last_reissued_at":"2026-05-18T01:25:11.061757Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:11.061757Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some relations between t(a,b,c,d;n) and N(a,b,c,d;n)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2015-11-19T14:15:42Z","abstract_excerpt":"Let $\\Bbb Z$ and $\\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\\in\\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$ $(x,y,z,w\\in\\Bbb Z$). 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