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]