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In contrast, we conjecture that any natural number is represented by $\\lfloor x^2/a\\rfloor+\\lfloor y^2/b\\rfloor +\\lfloor z^2/c\\rfloor$ with $x,y,z\\in\\mathbb Z$ if $(a,b,c)\\not=(1,1,1),(2,2,2)$, and that any natural number is represented by $\\lfloor T_x/a\\rfloor+\\lfloor T_y/b\\rfloor+\\lfloor T_z/c\\rfloor$ with $x,y,z\\in\\mathbb Z$, where $T_x$ denotes the triangular number $x(x+1)/2$. We confirm this general conjecture in some special cases"},"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":"1504.01608","kind":"arxiv","version":8},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-04-06T15:56:21Z","cross_cats_sorted":[],"title_canon_sha256":"7bbe4affb40e049853215476a05ff03a0dce53048955862c8a0a741e9d2f6cde","abstract_canon_sha256":"ed2b72e746bd98daf5e93301d78ff5232a71d6d170c8121d1e9a447afa47f963"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:50.736495Z","signature_b64":"Bs0cVy2dVhuk2tENjsN2TvSD5wp9sZZkpYiIzr0Ylzj2TBkycdspXWVdD4PPV/mJi58MA9kkTVeVWyi3LLOcBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3cf3a6c911c3e745dd98afc1068f6850f9ea5f40bebb586550966738dc09d3d5","last_reissued_at":"2026-05-18T01:23:50.735899Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:50.735899Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Natural numbers represented by $\\lfloor x^2/a\\rfloor+\\lfloor y^2/b\\rfloor+\\lfloor z^2/c\\rfloor$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2015-04-06T15:56:21Z","abstract_excerpt":"Let $a,b,c$ be positive integers. It is known that there are infinitely many positive integers not representated by $ax^2+by^2+cz^2$ with $x,y,z\\in\\mathbb Z$. In contrast, we conjecture that any natural number is represented by $\\lfloor x^2/a\\rfloor+\\lfloor y^2/b\\rfloor +\\lfloor z^2/c\\rfloor$ with $x,y,z\\in\\mathbb Z$ if $(a,b,c)\\not=(1,1,1),(2,2,2)$, and that any natural number is represented by $\\lfloor T_x/a\\rfloor+\\lfloor T_y/b\\rfloor+\\lfloor T_z/c\\rfloor$ with $x,y,z\\in\\mathbb Z$, where $T_x$ denotes the triangular number $x(x+1)/2$. 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