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A014602 Discriminants of imaginary quadratic fields with class number 1 (negated). 30
3, 4, 7, 8, 11, 19, 43, 67, 163 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Only fundamental discriminants are listed. The non-fundamental discriminants -12, -16, -27, and -28 also have class number 1 (and there are no others). - Andrew V. Sutherland, Apr 19 2009

REFERENCES

H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 229.

D. A. Cox, Primes of the form x^2+ny^2, Wiley, p. 271.

Siegel, Carl Ludwig. "Zum Beweise des Starkschen Satzes." Inventiones mathematicae 5.3 (1968): 180-191.

J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, see p. 483.

Stark, Harold M. "A complete determination of the complex quadratic fields of class-number one." The Michigan Mathematical Journal 14.1 (1967): 1-27.

LINKS

Table of n, a(n) for n=1..9.

A. Abatzoglou, A. Silverberg, A. V. Sutherland, A, Wong, A framework for deterministic primality proving using elliptic curves with complex multiplication, arXiv preprint arXiv:1404.0107, 2014.

Charles Delorme and Guillermo Pineda-Villavicencio, Quadratic Form Representations via Generalized Continuants, Journal of Integer Sequences, Vol. 18 (2015), Article 15.6.4.

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Eric Weisstein's World of Mathematics, Class Number

Index entries for sequences related to quadratic fields

MATHEMATICA

Union[ (-NumberFieldDiscriminant[ Sqrt[-#]] &) /@ Select[ Range[200], NumberFieldClassNumber[ Sqrt[-#]] == 1 &]] (* Jean-Fran├žois Alcover, Jan 04 2012 *)

PROG

(PARI) is(n)=isfundamental(-n) && qfbclassno(-n)==1 \\ Charles R Greathouse IV, Nov 20 2012

(Sage)

is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 1

A014602 = lambda n: filter(is_fund_and_qfbcn_1, (-1, -2, ..-n))

[-n for n in A014602(270)] # Peter Luschny, Aug 10 2014

CROSSREFS

Cf. A003656 (real case), A003173, A013658, A014603, A046002...A046020.

Sequence in context: A192051 A033195 A050032 * A078823 A045615 A211220

Adjacent sequences:  A014599 A014600 A014601 * A014603 A014604 A014605

KEYWORD

nonn,fini,full,nice

AUTHOR

Eric Rains (rains(AT)caltech.edu)

STATUS

approved

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Last modified December 9 14:23 EST 2016. Contains 278971 sequences.