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12, 21, 24, 28, 33, 44, 56, 57, 69, 76, 77, 88, 92, 93, 124, 129, 133, 141, 152, 161, 172, 177, 184, 188, 201, 209, 213, 217, 236, 237, 248, 249, 253, 268, 284, 301, 309, 329, 332, 341, 344, 376, 381, 393, 412, 413, 417, 428, 437, 453, 472, 489, 497, 501, 508, 517, 524, 536, 537, 553, 556, 573, 581, 589, 597
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OFFSET
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1,1
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COMMENTS
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Conjecture: if D is a term of this sequence, then D = uv, where u, v are 4, 8 or primes congruent to 3 modulo 4. For example, a(1) = 12 = 3*4, a(2) = 21 = 3*7, a(3) = 24 = 3*8, a(4) = 28 = 4*7, a(5) = 33 = 3*11, ... [This conjecture is correct: see Theorem 1 and Theorem 2 of Ezra Brown link; see also A003656. - Jianing Song, Dec 28 2021]
Let k be the quadratic field with discriminant D, O_k be ring of integers of k, N(x) be the norm of x and (D/p) be the Kronecker symbol. If D is a term of this sequence and D = uv, where u, v are 4, 8 or primes congruent to 3 modulo 4, then:
(a) if (((-u)/p), ((-v)/p)) = (1, 1), (1, 0) or (0, 1), then N(x) = p has solutions in O_k, while N(y) = -p has no solutions in k. For example, for D = 21 and p = 37, we have ((-3)/37) = ((-7)/37) = 1, and N(x) = 37 has solution x = (13 + sqrt(21))/2, but N(y) = -37 has no solutions in Q(sqrt(21)).
(b) if (((-u)/p), ((-v)/p)) = (-1, -1), (-1, 0) or (0, -1), then N(x) = -p has solutions in O_k, while N(y) = p has no solutions in k. For example, for D = 12 and p = 11, we have ((-3)/11) = ((-4)/11) = -1, and N(x) = -11 has solution x = 1 + 2*sqrt(3), but N(y) = 11 has no solutions in Q(sqrt(3)).
(c) if (((-u)/p), ((-v)/p)) = (1, -1) or (-1, 1), then N(x) = +-p has no solutions in k.
The smallest number of the form above that is not in this sequence is 316 = 4*79.
Also, it is conjectured that the quadratic field with discriminant D has form class number 2, where D is a term of this sequence. This is equivalent to the conjecture above. [This can also be deduced from the first paragraph of Ezra Brown link: the norm of the fundamental unit of the field k is -1 if D = 8 or a prime congruent to 1 modulo 4, and 1 if D is in this sequence. Here k is the quadratic field with discriminant D. - Jianing Song, Dec 28 2021]
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LINKS
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PROG
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(PARI) isA327297(D) = if(D>1&&isfundamental(D), quadclassunit(D)[1]==1&&!isprimepower(D), 0)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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