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tech / sci.math / Infinite Descent

Infinite Descent

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https://www.novabbs.com/tech/article-flat.php?id=133521&group=sci.math#133521

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NNTP-Posting-Date: Tue, 02 May 2023 19:03:54 +0000
From: amr...@net.inv (Feardul Amu)
Subject: Infinite Descent
Newsgroups: sci.math
Date: Tue, 2 May 2023 20:06:11 +0100
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by: Feardul Amu - Tue, 2 May 2023 19:06 UTC

I Think I understand infinite descent.

Given a triplet in cubes, there's always a smaller one.
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A triplet is three rational cubes................. no.1
A smaller triplet would be three rational cubes.... no.2

Since these are all rational cubes,
it would always be possible to get from no.1 to no.2
by manipulation of outer layers of the cubes in no.1

If there are no smaller triplets than the given one,
then No.2 is not rational.
Then there's no way to get from no.2 to no.1
by manipulation of layers of the cubes in no.1

Therefore No.1 cannot exist.

Therefore, given that there's one triplet
there's allways a smaller one, and an infinite number of them.

XYZ