From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: niu.tompaine Subject: Re: A good week of workshops, and a question. Date: 8 Oct 96 06:06:33 GMT Uh, Steve, you've got sort of a small collection of potential respondents here. I guess I'll have a go at it, but this is not going to be like reading the _Star_ (I hope). In article , karlson stephen h wrote: >Thursday and Friday, Nigel Boston of Illinois gave two talks on Fermat's >Last Theorem. I attended the general session Thursday, and it provoked a >question I didn't raise at workshop. Fermat proved that x^4 + y^4 = z^2 >has no solution in integers, and introduced a method of contradiction >called "infinite descent." The proof establishes the Last Theorem for any >even powers. Question: is Fermat's marginal note a reference to this >subsequently published result, with "truly marvelous" referring to the >infinite descent? > >Then, is his later work evidence that he recognized it doesn't work for >odd powers, particularly the odd primes? Have the mathematical >historians debated that possibility? Just in case anyone else is reading here, let me clarify: Fermat was well aware that two perfect squares could sum to a perfect square, but suspected that when you replace the word "square" with "cube" or "n-th power", there would be no solutions. This became a celebrated conjecture, proven in 1994. The situation was maddening in part because Fermat claimed to have a _proof_ that for all n > 2, the equation x^n + y^n = z^n had no solutions in positive integers x y z. (The "^" is for exponents.) (His claim was a pencilled-in comment in a book he was reading, and he said he didn't have room enough in the book margins to write the proof.) Fermat did leave a solution in the case n=4: he proved there were no positive-integer solutions to x^4 + y^4 = z^2. He did it by showing how he could take one solution to the equation shown and monkey around with the x y and z to get some other numbers a b and c say, with two nice features: a^4 + b^4 = c^2 and c < z. That is, you give him one solution and he'll give you another one which is smaller. Obviously this can't go on forever, the conclusion being that there couldn't have been any such solution x y z in the first place. It must have seemed pretty ingenious to Pierre, and maybe he thought some variant of that argument would work to show other equations can't have solutions, either. Many have suggested that something like this led him to think he had a proof that you could _never_ have two n-th powers sum to another n-th power for n bigger than 2. (for n=2 there are oodles of examples, starting with 3^2 + 4^2 = 5^2). Not wishing to discount completely the possibility that he really truly did have a proof along these lines, I have to side with the historians who think he made a little boo-boo here. With the benefit of a few centuries of hindsight we now know why infinite descent worked in his case: there's a lot of algebraic structure lurking in the solution set to x^4 = y^4 = z^2 (it's a mathematical group, as well as being topologically a donut) which just isn't there in the other cases. In fact, he almost didn't make it in the case n=4. This is hardly the forum for technical discussions, but take a good look at what he did: He wants to show two fourth powers (1, 16, 81, 256, ...) never sum to another fourth power. If he had tried infinite descent here, it would have failed, since (the necessary high-falutin' auxiliaries being absent) there is no way to take one such combination to give another one. What he thought to do turned out to be very lucky. He really showed that no two fourth powers can ever be a _square_ (1, 4, 9, 16, ...) That's _stronger_ than he needed to show (every fourth power is a perfect square but not vice versa). But it pays to try to prove the stronger claim in his case: if you could get two fourth powers to sum to a square, you could get two smaller ones also to sum to a _square_ (but not necessarily to a fourth power). Thus the reason Fermat succeeded in the case n=4 was because he was able to generalize the problem just a wee bit and so make it fall into a branch of number theory with particularly rich sstructure. Given what was known at that time I think it would be groundless adulation to think he knew that this was what he was doing; rather, he just noticed that it worked. Like I say, it's possible that in the more general case Fermat was able to _transform_ the problem into one where infinite descent would work, but as it stands, the Fermat problem is not amenable to infinite descent except (barely) in the case n=4. We now return you to your regularly-scheduled discussion group. dave