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A014602
Discriminants of imaginary quadratic fields with class number 1 (negated).
52
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
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
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
KEYWORD
nonn,fini,full,nice
AUTHOR
Eric Rains (rains(AT)caltech.edu)
STATUS
approved