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ON PERFECT $K$-RATIONAL CUBOIDS

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry Diophantine equations

Published online by Cambridge University Press:  04 October 2017

ANDREW BREMNER Open the ORCID record for ANDREW BREMNER [Opens in a new window]
ANDREW BREMNER*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe AZ 85287-1804, USA email [email protected]
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Abstract

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Let $K$ be an algebraic number field. A cuboid is said to be $K$-rational if its edges and face diagonals lie in $K$. A $K$-rational cuboid is said to be perfect if its body diagonal lies in $K$. The existence of perfect $\mathbb{Q}$-rational cuboids is an unsolved problem. We prove here that there are infinitely many distinct cubic fields $K$ such that a perfect $K$-rational cuboid exists; and that, for every integer $n\geq 2$, there is an algebraic number field $K$ of degree $n$ such that there exists a perfect $K$-rational cuboid.

Keywords

rational cuboidperfect cuboidcubic fieldRiemann–Roch

MSC classification

Primary: 11D41: Higher degree equations; Fermat's equation
Secondary: 11G05: Elliptic curves over global fields 11G35: Varieties over global fields 14G25: Global ground fields
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 1 , February 2018 , pp. 26 - 32
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

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