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A308420
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Squarefree numbers d of the form s^2 + r, where r divides 4s, such that Q(sqrt(d)) has class number 1.
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2
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2, 3, 5, 6, 7, 11, 13, 14, 17, 21, 23, 29, 33, 37, 38, 47, 53, 62, 69, 77, 83, 93, 101, 141, 167, 173, 197, 213, 227, 237, 293, 398, 413, 437, 453, 573, 677, 717, 1077, 1133, 1253, 1293, 1757
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OFFSET
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1,1
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COMMENTS
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This sequence is finite, but might not be given in full if the generalized Riemann hypothesis is false.
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REFERENCES
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Richard A. Mollin, Quadratics. p. 176, Theorem 5.4.3. Given "a fundamental discriminant of ERD-type with radicand D," the ring of Q(sqrt(D)) has class number 1 "if and only if D" is one of the values listed above, "with one possible exceptional value whose existence would be a counterexample to the GRH" (generalized Riemann hypothesis).
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LINKS
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FORMULA
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Given d = s^2 + r where r | 4s (this is called "extended Richaud-Degert type" or "ERD-type" by Mollin), d is in this sequence if h(O_Q(sqrt(d))) = 1, where h(O_K) is the class number of the ring of algebraic integers O_K.
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EXAMPLE
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Since 7 = 3^2 - 2 (note that 2 is a divisor of 4 * 9) and h(Z[sqrt(7)]) = 1, 7 is in the sequence.
Although 10 = 3^2 + 1, we see that h(Z[sqrt(10)]) > 1 since Z[sqrt(10)] is not a unique factorization domain (e.g., 10 = 2 * 5 = sqrt(10)^2). So 10 is not in the sequence.
Although h(Z[sqrt(19)]) = 1, there is no way to express 19 as s^2 + r, e.g., 19 = 3^2 + 10 but 10 is not a divisor of 12, 19 = 4^2 + 3 but 3 is not a divisor of 16, 19 = 5^2 - 6 but 6 is not a divisor of 20, 19 = 6^2 - 17 but -17 does not divide 24. So 19 is not in the sequence either.
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CROSSREFS
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KEYWORD
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nonn,fini
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AUTHOR
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STATUS
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approved
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