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For every n=4,5,... we prove that the least prime p>3n with 3|p-1 is just the least positive integer m such that 18k(3k-1) (k=1,...,n) are pairwise distinct modulo m. For d=4,6,12 and n=3,4,...., we prove that the least prime p>2n-2 with p=-1 (mod d) is the smallest integer m such that those (2k-1)^d for k=1,...,n are pairwise distinc"},"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":"1202.6589","kind":"arxiv","version":16},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-02-29T16:20:47Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"04d181380e67934033e39bdaf107086a90dd0f65a15285a846209c8746b06cea","abstract_canon_sha256":"9fe774d775eaf817b04a860e858296e5d00cb04d26103922f15bba430b2bcc24"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:27:52.424988Z","signature_b64":"7+D16YqqnUupCvuaOY89EHiCIvSbWJUuFw3qA9nJ4gooKxQqXsJ6NJtqGyQVetoTcXVv7h6JYSfv1WKInz+ZDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de51d8b627fab7cf981943c785e1df0dd4cd6286f0ed8a63350a04170af099ca","last_reissued_at":"2026-05-18T03:27:52.424193Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:27:52.424193Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On functions taking only prime values","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2012-02-29T16:20:47Z","abstract_excerpt":"For n=1,2,3,... define S(n) as the smallest integer m>1 such that those 2k(k-1) mod m for k=1,...,n are pairwise distinct; we show that S(n) is the least prime greater than 2n-2 and hence the value set of the function S(n) is exactly the set of all prime numbers. 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