### Abstract

Let (Formula presented.) be an algebraic number field. A cuboid is said to be (Formula presented.)-rational if its edges and face diagonals lie in (Formula presented.). A (Formula presented.)-rational cuboid is said to be perfect if its body diagonal lies in (Formula presented.). The existence of perfect (Formula presented.)-rational cuboids is an unsolved problem. We prove here that there are infinitely many distinct cubic fields (Formula presented.) such that a perfect (Formula presented.)-rational cuboid exists; and that, for every integer (Formula presented.), there is an algebraic number field (Formula presented.) of degree (Formula presented.) such that there exists a perfect (Formula presented.)-rational cuboid.

Original language | English (US) |
---|---|

Pages (from-to) | 1-7 |

Number of pages | 7 |

Journal | Bulletin of the Australian Mathematical Society |

DOIs | |

State | Accepted/In press - Oct 4 2017 |

### Fingerprint

### Keywords

- cubic field
- perfect cuboid
- rational cuboid
- Riemann–Roch

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**ON PERFECT K-RATIONAL CUBOIDS.** / Bremner, Andrew.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - ON PERFECT K-RATIONAL CUBOIDS

AU - Bremner, Andrew

PY - 2017/10/4

Y1 - 2017/10/4

N2 - Let (Formula presented.) be an algebraic number field. A cuboid is said to be (Formula presented.)-rational if its edges and face diagonals lie in (Formula presented.). A (Formula presented.)-rational cuboid is said to be perfect if its body diagonal lies in (Formula presented.). The existence of perfect (Formula presented.)-rational cuboids is an unsolved problem. We prove here that there are infinitely many distinct cubic fields (Formula presented.) such that a perfect (Formula presented.)-rational cuboid exists; and that, for every integer (Formula presented.), there is an algebraic number field (Formula presented.) of degree (Formula presented.) such that there exists a perfect (Formula presented.)-rational cuboid.

AB - Let (Formula presented.) be an algebraic number field. A cuboid is said to be (Formula presented.)-rational if its edges and face diagonals lie in (Formula presented.). A (Formula presented.)-rational cuboid is said to be perfect if its body diagonal lies in (Formula presented.). The existence of perfect (Formula presented.)-rational cuboids is an unsolved problem. We prove here that there are infinitely many distinct cubic fields (Formula presented.) such that a perfect (Formula presented.)-rational cuboid exists; and that, for every integer (Formula presented.), there is an algebraic number field (Formula presented.) of degree (Formula presented.) such that there exists a perfect (Formula presented.)-rational cuboid.

KW - cubic field

KW - perfect cuboid

KW - rational cuboid

KW - Riemann–Roch

UR - http://www.scopus.com/inward/record.url?scp=85030867568&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85030867568&partnerID=8YFLogxK

U2 - 10.1017/S0004972717000697

DO - 10.1017/S0004972717000697

M3 - Article

AN - SCOPUS:85030867568

SP - 1

EP - 7

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

SN - 0004-9727

ER -