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

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

Offset: 1

Author

Eric Rains (rains(AT)caltech.edu)

Keywords

• nonn
• fini
• full
• nice

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

• 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

Crossrefs

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

Programs

Mathematica

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

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