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ON PERFECT $K$-RATIONAL CUBOIDS

Published online by Cambridge University Press:  04 October 2017

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.

Type
Research Article
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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