A006203 Discriminants of imaginary quadratic fields with class number 3 (negated).

23, 31, 59, 83, 107, 139, 211, 283, 307, 331, 379, 499, 547, 643, 883, 907

Offset: 1

Author

N. J. A. Sloane

Keywords

• fini
• nonn
• full
• nice

Comments

Also n such that Q(sqrt(-n)) has class number 3. Lubelski in 1936 proved that 907 is maximal term of this sequence. - Artur Jasinski, Oct 07 2011

References

• H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 514.
• J. M. Masley, Where are the number fields with small class number?, pp. 221-242 of Number Theory Carbondale 1979, Lect. Notes Math. 751 (1982).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Steven Arno, M. L. Robinson, Ferrell S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arith. 83 (1998), pp. 295-330.
• Kurt Heegner, Diophantische Analysis und Modulfunktionen, Matematische Zaitschrift, 1952, Vol. 56. p. 253.
• S. Lubelski, Zur Reduzibilität von Polynomen in Kongruenzentheorie, Acta Arithmetica 1 (1935) pp. 169-183.
• Keith Matthews, Tables of imaginary quadratic fields with small class numbers
• Pieter Moree and Armand Noubissie, Higher Reciprocity Laws and Ternary Linear Recurrence Sequences, arXiv:2205.06685 [math.NT], 2022. See p. 4.
• Eric Weisstein's World of Mathematics, Class Number.
• Index entries for sequences related to quadratic fields

Crossrefs

Cf. A005847, A013658, A014602, A014603, A046002, ..., A046020.

Cf. also A003173, A005847, ...

Cf. A191410.

Programs

Mathematica

Union[ (-NumberFieldDiscriminant[ Sqrt[-#]] & ) /@ Select[ Range[1000], NumberFieldClassNumber[ Sqrt[-#]] == 3 & ]] (* Jean-François Alcover, Jan 04 2012 *)

PARI

ok(n)={isfundamental(-n) && quadclassunit(-n).no == 3} \\ Andrew Howroyd, Jul 20 2018

Sage

[n for n in (1..1000) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==3] # G. C. Greubel, Mar 01 2019