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A355461
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Squarefree numbers d of the form r^2*m^2 + 4*r, where r and m are odd positive integers, such that Q(sqrt(d)) has class number 1.
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0
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5, 13, 21, 29, 53, 173, 237, 293, 437, 453, 1133, 1253
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OFFSET
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1,1
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COMMENTS
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In 1801, Gauss conjectured that there exist infinitely many real quadratic fields with class number one and the conjecture is still unproved, but there are only 12 real quadratic fields of class number one which are of the form Q(sqrt(r^2*m^2 + 4*r)), where the parameters r and m are odd integers. Those 12 values of d := r^2*m^2 + 4*r belong to the present sequence.
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LINKS
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Table of n, a(n) for n=1..12.
A. Biró and K. Lapkova The class number one problem for the real quadratic fields Q(sqrt(a*n^2+4*a)), Acta Arith., vol. 172(2), 2016, pp. 117-131.
A. Hoque and S. Kotyada Class number one problem for the real quadratic fields Q(sqrt(m^2+2*r)), Archiv der Mathematik, vol. 116(1), 2021, pp. 33-36.
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EXAMPLE
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a(2) = 13 since h(13) = h(1^2*3^2 + 4*1) = 1.
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CROSSREFS
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Cf. A050950, A053329, A308420.
Sequence in context: A191155 A107996 A107997 * A166095 A166090 A065766
Adjacent sequences: A355458 A355459 A355460 * A355462 A355463 A355464
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KEYWORD
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nonn,fini,full
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AUTHOR
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Marco Ripà, Jul 02 2022
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STATUS
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approved
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