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A244574
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Absolute discriminants of complex quadratic fields with 3-class rank 3 and thus with infinite class tower.
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1
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3321607, 3640387, 4019207, 4447704, 4472360, 4818916, 4897363, 5048347, 5067967, 5153431, 5288968, 5769988, 6562327, 7016747, 7060148, 7503391, 7546164, 8124503, 8180671, 8721735, 8819519, 8992363, 9379703, 9487991, 9778603
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OFFSET
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1,1
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COMMENTS
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Diaz y Diaz discovered a(1), a(2) and three other terms in 1973. However, Buell was the first who proved minimality of a(1). According to Koch and Venkov, 3-class rank 3 ensures an infinite Hilbert (3-)class field tower.
The first 25 terms were computed with MAGMA over 18 hours of CPU time.
With exception of a(16)=7503391, all terms below 10^7 and lots of further terms below 10^8 are given in Appendice 1, pp. 66-77, of the Thesis of F. Diaz y Diaz (1978). - Daniel Constantin Mayer, Sep 27 2014
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REFERENCES
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F. Diaz y Diaz, Sur le 3-rang des corps quadratiques, Publ. math. d'Orsay, No. 78-11, Univ. Paris-Sud (1978).
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LINKS
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Table of n, a(n) for n=1..25.
D. A. Buell, Class groups of quadratic fields, Math. Comp. 30 (1976), no. 135, 610-623.
Francisco Diaz y Diaz, Sur les corps quadratiques imaginaires dont le 3-rang du groupe des classes est supérieur à 1, Séminaire Delange-Pisot-Poitou, 1973/74, no. G15.
H. Koch, B. B. Venkov, Über den p-Klassenkörperturm eines imaginär-quadratischen Zahlkörpers, Astérisque 24-25 (1975), 57-67.
D. C. Mayer, Complex quadratic fields of type (3, 3, 3), 2014.
Daniel C. Mayer, Index-p abelianization data of p-class tower groups, arXiv preprint arXiv:1502.03388, 2015
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EXAMPLE
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3-class group of type (9,3,3) for a(1)=3321607, and of type (3,3,3) for a(4)=4447704. Unique 3-class group of type (27,3,3) for a(10)=5153431.
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PROG
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(MAGMA) for d := 1 to 10^7 do a := false; if (3 eq d mod 4) and IsSquarefree(d) then a := true; end if; if (0 eq d mod 4) then r := d div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (1 eq r mod 4)) then a := true; end if; end if; if (true eq a) then K := QuadraticField(-d); C := ClassGroup(K); if (3 eq #pPrimaryInvariants(C, 3)) then d, ", "; end if; end if; end for;
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CROSSREFS
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Cf. A242862, A244575 (a subsequence).
Sequence in context: A206316 A186959 A186594 * A250675 A242608 A206511
Adjacent sequences: A244571 A244572 A244573 * A244575 A244576 A244577
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KEYWORD
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hard,more,nonn
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AUTHOR
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Daniel Constantin Mayer, Jun 30 2014
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STATUS
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approved
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