A014602 Discriminants of imaginary quadratic fields with class number 1 (negated).

3, 4, 7, 8, 11, 19, 43, 67, 163

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.

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

Links

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

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

Giacomo Cherubini and Alessandro Fazzari, Hyperbolic angles from Heegner points, arXiv:2206.08282 [math.NT], 2022. Mentions this sequence.

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

Erich Kaltofen and Heinrich Rolletschek, Computing greatest common divisors and factorizations in quadratic number fields, Mathematics of Computation 53.188 (1989): 697-720. See page 698.

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

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

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

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-François 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: A050032 A330221 A343151 * 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