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For every $n \\ge 3$ we give appropriate systems of linear forms (or equivalently basis) describing all $n \\times n$ magic squares with integer entries and we calculate the complexity of these systems in the Green and Tao sense. We compute the precise asymptotics for the cases $n=3$ (complexity 3) and $n=4$ (complexity 1), and the given algorithm works for $n \\ge 5$ (complexity 1). Finally, we show that the asymptoti"},"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":"1207.3936","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-07-17T10:34:58Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"3aa8a20c6cd1e7b5098dc35dc7635954b9be56061c31e185690854059d3cf821","abstract_canon_sha256":"bd817399fbf5fb0832f6cd59456da3716ac38949548bb12533e9fe249702f639"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:49.457176Z","signature_b64":"5wiD15QaSpY6lWdboX6vV0337peLkK4AZOcBUGo92BX8JQCFosx7pRwl90Hgndb4hHa67EbPOo0fA+JTZsvEDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c6e18746ef09c560b0846e412957859b46fe202e212fb60844e03e47218657b","last_reissued_at":"2026-05-18T03:50:49.455860Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:49.455860Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotics for Magic Squares of Primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Carlos Vinuesa","submitted_at":"2012-07-17T10:34:58Z","abstract_excerpt":"Based on the work of Green, Tao and Ziegler, we give asymptotics when $N \\to \\infty$ for the number of $n \\times n$ magic squares with their entries being prime numbers in $[0,N]$. For every $n \\ge 3$ we give appropriate systems of linear forms (or equivalently basis) describing all $n \\times n$ magic squares with integer entries and we calculate the complexity of these systems in the Green and Tao sense. We compute the precise asymptotics for the cases $n=3$ (complexity 3) and $n=4$ (complexity 1), and the given algorithm works for $n \\ge 5$ (complexity 1). 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