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arXiv · 2004.06039

Reducing radicals in the spirit of Euclid

Kurt Girstmair
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Abstract

Let $p$ be an odd natural number $\ge 3$. Inspired by results from Euclid's {\em Elements}, we express the irrational $$y=\sqrt[p]{d+\sqrt R}, $$ whose degree is $2p$, as a polynomial function of irrationals of degrees $\le p$. In certain cases $y$ is expressed by simple radicals. This reduction of the degree exhibits remarkably regular patterns of the polynomials involved. The proof is based on hypergeometric summation, in particular, on Zeilberger's algorithm.

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Kurt Girstmair. 2020-04-13. Reducing radicals in the spirit of Euclid. https://arxiv.org/abs/2004.06039

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