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Furthermore, we present several infinite families of integers b such that the D(-1) pair {1, b} cannot be extended to a D(-1) quadruple. For instance, we show that if r=5p where p is an odd prime, then the D(-1) pair {1, r^2+1} cannot be extended to a D(-1) quadruple."},"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":"1309.4347","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-17T15:18:03Z","cross_cats_sorted":[],"title_canon_sha256":"b24f8e05ef2c28e36229bda3fc8f435b5f886cce2f0e543fe9f6971cb5f155e6","abstract_canon_sha256":"51a4ea562e1f196ed983bacf488de024b5e2478ad113719a0b67526d8f44883b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:09.016496Z","signature_b64":"8Xbly4D3Ht4MlRyyVIQ4lg46I+HqaJw/Vnj6Hu+D51HqQoKjqFRMB+PbjKpytkEUCxOCZYc8seL1AB/1XP3KCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d6f60b2b47ee3fc7803dae16ba02655d202f3eef73e765adbd61fd80d2672e48","last_reissued_at":"2026-05-18T03:13:09.015653Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:09.015653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the prime divisors of elements of a $D(-1)$ quadruple","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Anitha Srinivasan","submitted_at":"2013-09-17T15:18:03Z","abstract_excerpt":"We show that if {1, b, c, d} is a D(-1) diophantine quadruple with b<c<d and c=1+s^2, then the cases s=p^k, s=2p^k, c=p and c=2p^k do not occur, where p is an odd prime and k is a positive integer. For the integer d=1+x^2, we show that it is not prime and that x is divisible by at least two distinct odd primes. Furthermore, we present several infinite families of integers b such that the D(-1) pair {1, b} cannot be extended to a D(-1) quadruple. 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