[seqfan] Re: Perfectly Amicable Boxes

[Chris Thompson]

On Jan 29 2019, Hans Havermann wrote:

>Pat Ballew is looking for Perfectly Amicable Boxes:
>
>https://pballew.blogspot.com/2019/01/looking-for-perfectly-amicable-boxes.html
>
>My guess is that there aren't any but perhaps someone here can demonstrate
>that. In the meantime I was looking at A103277 which is relevant to the
>subject. I couldn't parse the Mathematica code for that sequence. Perhaps
>that can be fixed.

I take it that *Perfectly* Amicable Boxes refers to the question "are there pairs, or triples of Boxes which share the same total edge lengths, surface
area, and volume". In which case, no there aren't any.

If the box is a X b X c, specifying the edge lengths fixes a+b+c, the
surface ab+bc+ca, and the volume abc. Therefore you have fixed the cubic
(x-a)(x-b)(x-c) = x^3 - (a+b+c)x^2 + (ab+bc+ca)x - abc, and therefore its
three roots, up to a permutation.

Or to put it another way, if you have fixed all the elementary symmetric
polynomials of a set of numbers, you have fixed the numbers themselves,
up to a permutation.

-- 
Chris Thompson
Email: cet1 at cam.ac.uk