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We show that for every $n$ of the form $4^k+1$, $T(n)$ and $Q(n)$ are both at least $n^{\\Omega(n)}$. This result confirms a conjecture of Rivin, Vardi and Zimmerman for these values of $n$. We also present new upper bounds on $T(n)$ and $Q(n)$ using the entropy method, and conjecture that in the case of $T(n)$ the bound "},"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":"1705.05225","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-15T13:34:59Z","cross_cats_sorted":[],"title_canon_sha256":"546fa61e833e0d7442ae8d13e1a562dd01181a093d963b693794218908d1e75c","abstract_canon_sha256":"0fb01248175960374553d791195b883a0231562b65262ebcd423079b47e60c68"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:22.864055Z","signature_b64":"tGh3s3fiXbsu/6LIx19jPDb+ExAUr7RVE5MyXPpkjpGokTLvSSTZIBTAgWDhtFmjt78MUw4GgXTWTa1wlvwWAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"245092b81e87ea40808550ba99b0a0806ddaba75aef1359394a8ecd04547e7f6","last_reissued_at":"2026-05-18T00:44:22.863431Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:22.863431Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New bounds on the number of n-queens configurations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Zur Luria","submitted_at":"2017-05-15T13:34:59Z","abstract_excerpt":"In how many ways can $n$ queens be placed on an $n \\times n$ chessboard so that no two queens attack each other? 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