From: richard.wheeler@pop.net.ntl.com
Subject: Re: Update
Date: Sun, 03 Oct 1999 09:59:05 +0100
Newsgroups: sci.math
Keywords: What is the method of infinite descent?

James Harris wrote:

> Keith Ramsay wrote in message
> > Such a proof would not normally be called a proof by infinite descent.
> > At the beginning of the argument, one can replace the hypothetical
> > counterexample with a primitive counterexample (i.e., in which x, y,
> > and z have no common factor >1). That's not usually thought of in
> > terms of descending over and over.
> >
> > Infinite descent ordinarily is used to describe a proof such as
> > Fermat's proof of the p=4 case. On the assumption that a
> > counterexample exists, he proves that a counterexample having a
> > smaller value of z would also exist, and not by demonstrating that
> > x, y, and z have a common factor. The existence of a solution and of
> > a smaller solution doesn't immediately yield a contradiction. On the
> > other hand, by the same argument, "repeated", there would have to
> > exist an infinite sequence of solutions having descending values of z.
> > That's impossible, hence the term "infinite descent".
> >
> You're thinking too narrowly in your definition.  Fermat didn't use his
> infinite descent method with FLT alone.  For all we know, he used it on
> other proofs first.
>
> So, again we have one of those discussions where you guys are using what you
> know from today's usage and I'm talking from a perspective that spans
> hundreds of years.
>
> James Harris

 No Keith's definition is exactly right, although Fermat probably wouldn't
have been familiar with the projective line aspect that Keith mentioned.

As used by Fermat, for FLT cases and many other problems, the method
of descent starts with one (or more) equations, and then from these via
arithmetic reasoning derives equation[s] identical in form to the first but
for which the size of the unknowns is in some sense less than the first.

This measure of size is often called the "height" (although that term has a
technical definition in other areas such as quatratic forms), and is defined
in such a way that it is always positive (or possibly zero for trivial
solutions).

One can define the height in any way that is convenient (i.e. easy to prove
the reduction in height). For example it could be just the absolute value of
one of the unknowns, but more often the product of two or more, perhaps
all, is easier to use.

The Method of Descent is most commonly used with homogenous
equations, for which multiplying (or dividing) every unknown by the
same factor leaves the equations unchanged.

Dividing out common factors in homogenous equations at the start
is not directly part of the descent but is just an initial one-off tidy up
operation, like rolling up ones sleeves before getting down to the
nitty gritty, although it is usually necessary in order to allow the
descent argument to work and in that sense is part of the package
so to speak.

To use descent to prove the equation[s] impossible, one starts by
assuming a solution with a non-zero height, and then one shows that
in the derived system, with the smaller height, the height is _still_
non-zero. That is the key point, because it implies an infinite sequence
of non-zero positive integers each strictly smaller than the preceding
integer, which of course starting from some finite integer is impossible.

An equivalent formulation is to start by assuming that one has a solution
for which the height is minimal, and then the derived solution with smaller
height is in itself a contradiction.

There are various other points one could make about the Method of
Descent. In some problems one has to juggle with more than one set
of equations, for example a starting set A might imply either A (with
smaller height) or B, and B could imply B with smaller height, in which
case A (and B) are both impossible.

Sometimes, more interestingly, solutions can slip through the net.
In the preceding example B might imply A or B or C, and C may
turn out to have a solution after all. In that case the descent via A
and B can then be reversed to become a conduit for generating
every other solution.

Also, some systems exhibit what might be called "imperfect
descent", where the descent argument only applies if the height
exceeds some minimum non-zero value. In that case there may
be non-trivial solutions with height less than that value, and again
the set of these solutions generate all other solutions by working
back up the ladder.

Cheers

John R Ramsden  (jr@redmink.demon.co.uk)

[posted under an alias]