A133315 Positive discriminants of orders of quadratic fields with class number 1.

5, 8, 12, 13, 17, 20, 21, 24, 28, 29, 32, 33, 37, 41, 44, 45, 48, 52, 53, 56, 57, 61, 68, 69, 72, 73, 76, 77, 80, 84, 88, 89, 92, 93, 97, 101, 108, 109, 112, 113, 116, 117, 124, 125, 128, 129, 132, 133, 137, 141, 149, 152, 153, 157, 161, 164, 172, 173, 176, 177, 180, 181, 184, 188, 189, 193, 197, 201, 208

Offset

1,1

Comments

Not to be confused with A391423, the positive discriminants of orders of quadratic fields with *form* class number 1. - Jianing Song, Dec 09 2025

Discriminants D of orders of real quadratic fields such that every binary quadratic form of discriminant D is equivalent to f_0 or -f_0, where f_0 is the principal form. (So f_0 corresponds to the form x^2 - (D/4)*y^2 for 4|D and x^2 - x*y - ((D-1)/4)*y^2 for D == 1 (mod 4), and -f_0 corresponds to the form (D/4)*x^2 - y^2 for 4|D and ((D-1)/4)*x^2 - x*y - y^2 for D == 1 (mod 4)). - Jianing Song, Dec 09 2025

References

D. A. Cox, Primes of the form x^2+ny^2, Wiley, New York, 1989.

D. E. Flath, Introduction to Number Theory, Wiley-Interscience, 1989.

Links

Jianing Song, Table of n, a(n) for n = 1..10000

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

Prog

(PARI) isA133315 (n) = (n % 4 <= 1) && (! issquare(n)) && (qfbclassno(n, 1) == 1) \\ Michel Marcus, May 23 2013

Crossrefs

Sequences related to the class numbers of real quadratic fields:

| Class numbers | Form class no. |

-------------+---------------+----------------+

Fundamental | 1: A003656 | 1: A003655 |

disc. only | 2: A094619 | 2: A391420 |

(A003658) | 3: A094612 | 3: A391421 |

| List: A003652 | List: A003646 |

-------------+---------------+----------------+

All disc. | 1: this seq. | 1: A391423 |

(A079896) | 2: A344408 | 2: A391424 |

| 3: A344409 | 3: A391425 |

| List: A391418 | List: A087048 |

For a list of sequences related to the class groups of real quadratic fields, see A390079.

Sequence in context: A079896 A391419 A390079 * A003658 A391417 A391422

Adjacent sequences: A133312 A133313 A133314 * A133316 A133317 A133318

Keyword

nonn

Author

David Brink, Dec 30 2007

Extensions

Name clarified by Jianing Song, Dec 09 2025

Status

approved