From: bumby@lagrange.rutgers.edu (Richard Bumby)
Subject: Re: some radical identities
Date: 16 Aug 1999 17:06:59 -0400
Newsgroups: sci.math
Keywords: Keywords: Some nested radical expressions simplify
bruck@math.usc.edu (Ronald Bruck) writes:
>In article <7p335k$6au$1@nnrp1.deja.com>, David Bernier
> wrote:
>:Years ago, I read somewhere about the identity:
>: sqrt(5)+sqrt(22+2*sqrt(5))
>:= sqrt(11+2*sqrt(29))+sqrt(16-2*sqrt(29)+2*sqrt(55-10*sqrt(29))) (1)
>:
>:It may have been in Scientific American; I checked the simpler
>:identity 1+sqrt(3)=sqrt(3+sqrt(13+4*sqrt(3))) (Maple can simplify the
>:right hand side here).
I remember that identity from an article called "Incredible
Identities" by Dan Shanks in volume 12 of the Fibonacci Quarterly.
The year was 1974. I was reminded of it several years later while
attending a lecture on Book X of Euclid's "Elements", and was able to
publish my observations as "Incredible Identities Revisited" in volume
25 of the Fibonacci Quarterly. That year was 1987. Since Usenet lacks
the publication lag of print journals, current interest seems to be a
sign of a 13 year period in this subject.
>:
>:The version of Maple at the Inverse Symbolic Calculator site
>:did not reduce the "RHS" of (1) to its "LHS". A candidate for
>:the irreducible polynomial of sqrt(5)+sqrt(22+2*sqrt(5)) is
>: x^4 - 54*x^2 - 40*x +269.
>:
This is not the whole story. Showing that both expressions satisfy
this polynomial only shows that they are algebraically conjugate. It
takes a little more to show that they are equal.
>:I presume there is a method to finding an identity like (1), and
>:I'm curious to know about similar algebraic identities.
See the "additional reference" mentioned in my paper -- an article by
Shanks that explains how he was led to the original identity.
>I remember reading (it's been a few years now) about an algorithm which
>simplifies radical expressions like this. But I can't remember where;
>maybe the Notices of the AMS? It wasn't a mathematics paper, it was some
>sort of news article.
>But the algorithm, while completely solving the problem, was prohibitively
>expensive in time. No explanation of what that meant.
>Can anyone fill in the details? The only thing I can remember is that the
>mathematician was an algebraist, and female. (Well, that cuts it down to,
>what, maybe 5% of the world's mathematicians?)
Since others have identified the author as Susan Landau, the Reviews
in Number Theory tells me that the paper you are thinking of must be
"How to tangle with a nested radical" in volume 16 (1994) of the
Math. Intelligencer.
>--Ron Bruck
--
R. T. Bumby ** Rutgers Math || Amer. Math. Monthly Problems Editor 1992--1996
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