### 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) |
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Pages (from-to) | 1-7 |

Number of pages | 7 |

Journal | Bulletin of the Australian Mathematical Society |

DOIs | |

State | Accepted/In press - Oct 4 2017 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)