A003644 Discriminants of the known imaginary quadratic fields with 1 class per genus (a finite sequence).

3, 4, 7, 8, 11, 15, 19, 20, 24, 35, 40, 43, 51, 52, 67, 84, 88, 91, 115, 120, 123, 132, 148, 163, 168, 187, 195, 228, 232, 235, 267, 280, 312, 340, 372, 403, 408, 420, 427, 435, 483, 520, 532, 555, 595, 627, 660, 708, 715, 760, 795, 840, 1012, 1092, 1155, 1320, 1380, 1428, 1435, 1540, 1848, 1995, 3003, 3315, 5460

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• fini
• full
• nice

Comments

This is the complete table from Borevich and Shafarevich.

If the GRH is true, the list contains the discriminants of all imaginary quadratic fields with 1 class per genus; otherwise, there may be one more such discriminant not on the list. (See Weinberger.) - Everett W. Howe, Aug 01 2014

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425-430.
• L. E. Dickson, Introduction to the Theory of Numbers. Dover, NY, 1957, p. 85.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Shin-ichi Katayama, Rational Points on the Parabola and the Arithmetic of Related Algebraic Tori, J. Math. Tokushima Univ. (2024) Vol. 58, 11-31. See p. 29.
• Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
• P. J. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith. 22 (1973), 117-124.

Crossrefs

Cf. A133288, A316743.

Programs

PARI

ok(n)={isfundamental(-n) && !#select(k->k<>2, quadclassunit(-n).cyc)} \\ Andrew Howroyd, Jul 20 2018

Extensions

Clarified name (added "the known") - Everett W. Howe, Aug 01 2014