### Abstract

Ono's number p_{D} and the class number h_{D}, associated to an imaginary quadratic field with discriminant -D, are closely connected. For example, Frobenius-Rabinowitsch Theorem asserts that p_{D} = 1 if and only if h_{D} = 1. In 1986, T. Ono raised a problem whether the inequality h_{D} ≤ 2^{pD} holds. However, in our previous paper [8], we saw that there are infinitely many D such that the inequality does not hold. In this paper we give a modification to the inequality h_{D} ≤ 2^{pD}. We also discuss lower and upper bounds for Ono's number p_{D}.

Original language | English |
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Pages (from-to) | 105-108 |

Number of pages | 4 |

Journal | Proceedings of the Japan Academy Series A: Mathematical Sciences |

Volume | 78 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2002 Jan 1 |

Externally published | Yes |

### Keywords

- Class number
- Ono's number

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**An inequality between class numbers and Ono's numbers associated to imaginary quadratic fields.** / Sairaiji, Fumio; Shimizu, Kenichi.

Research output: Contribution to journal › Article

*Proceedings of the Japan Academy Series A: Mathematical Sciences*, vol. 78, no. 7, pp. 105-108. https://doi.org/10.3792/pjaa.78.105

}

TY - JOUR

T1 - An inequality between class numbers and Ono's numbers associated to imaginary quadratic fields

AU - Sairaiji, Fumio

AU - Shimizu, Kenichi

PY - 2002/1/1

Y1 - 2002/1/1

N2 - Ono's number pD and the class number hD, associated to an imaginary quadratic field with discriminant -D, are closely connected. For example, Frobenius-Rabinowitsch Theorem asserts that pD = 1 if and only if hD = 1. In 1986, T. Ono raised a problem whether the inequality hD ≤ 2pD holds. However, in our previous paper [8], we saw that there are infinitely many D such that the inequality does not hold. In this paper we give a modification to the inequality hD ≤ 2pD. We also discuss lower and upper bounds for Ono's number pD.

AB - Ono's number pD and the class number hD, associated to an imaginary quadratic field with discriminant -D, are closely connected. For example, Frobenius-Rabinowitsch Theorem asserts that pD = 1 if and only if hD = 1. In 1986, T. Ono raised a problem whether the inequality hD ≤ 2pD holds. However, in our previous paper [8], we saw that there are infinitely many D such that the inequality does not hold. In this paper we give a modification to the inequality hD ≤ 2pD. We also discuss lower and upper bounds for Ono's number pD.

KW - Class number

KW - Ono's number

UR - http://www.scopus.com/inward/record.url?scp=0036765147&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036765147&partnerID=8YFLogxK

U2 - 10.3792/pjaa.78.105

DO - 10.3792/pjaa.78.105

M3 - Article

VL - 78

SP - 105

EP - 108

JO - Proceedings of the Japan Academy Series A: Mathematical Sciences

JF - Proceedings of the Japan Academy Series A: Mathematical Sciences

SN - 0386-2194

IS - 7

ER -