In fact, Fermat himself may very well have later realized that he was mistaken, but seen no need to go back and correct a marginal note-to-himself.
I agree that it would be fascinating to find a valid proof that could have been Fermat’s. It would mean that there’s some trick or technique that mathematicians have been missing for hundreds of years. But I’m not holding my breath.
Yeah. If I had to bet, I’d say that Fermat found one of the beautiful and elegant but flawed proofs that were advanced in the years after his passing.
Seems I was under estimating the complexity of the protein folding problem in my earlier post. To predict how one protein folds, it takes 10,000 CPU days, or 30 CPU years!
Since the Four-Color Theorem was discussed above, it’s worth pointing out that today’s “Straight Dope Classic” is a re-run of Cecil’s old column on the issue, and that there’s a thread about it in the Comment on Cecil’s Columns forum, in which Post #5, by John W. Kennedy, is particularly enlightening.
Cecil himself is a bit less so. He said that “Basically what the computer did was check out all the possible map combinations by trial and error,” which is an oversimplification, and that “There are those who complain that this process does not constitute a mathematical proof, as that term is usually understood, but rather falls more into the category of an experiment, understandably something of a novelty in the field of abstract mathematics,” which I don’t really understand and think is probably misleading.
He did in fact come up with a proof for the n=3 and n=4 cases, and he probably thought, at the time he wrote his margin note, that his method would generalize to all n.
And point taken on having to not only find an algorithm, but to prove that it works… That could certainly be nontrivial.
I have a MA in Mathematics. I am interested in Number Theory (although haven’t taken a course beyond the basics). I don’t have a frickin’ clue what the hell is going on with the Riemann Zeta function and every attempt to try to understand it makes me more hopelessly confused.
However.
All that I’ve learned about the conjecture involving said function is that it is at the very heart of modern number theoretic concerns, so much so that (I was told anyway) there are some propositions that were proved by proving it was true if the GRH was true and it was true if the GRH was false. I have never encountered such a deep result anywhere else, and it’s been in the way of elegant proofs for quite a long time. However, it seems simple enough and quite unlike the continuum hypothesis (proven to be independent of the rest of set theory) that it should be solvable one way or another.
Goldbach’s conjecture is really the next Fermat’s Last theorem, if without the whole intrigue behind the supposed proof its conjecturer may have had. Assuming it requires some advanced machinery to solve, it’ll probably end up as a dribbling little corollary to some big important result like Fermat’s was to the Taniyama–Shimura conjecture; that result is rather “beautiful” and interesting in itself, but work on its solution ended up being really motivated by someone figuring out it would allow a simple proof to some inconsequential result that for no particularly good reason people were interested in. The full strength modularity theorem (what the T-S conjecture is now known as) is able to prove a whole bunch of very similar results which would probabaly be just as interesting if someone had claimed they had proven them 400 years ago.
John Derbyshire’s book Prime Obsession is the best explanation of the Riemann Hypothesis at a level that an intelligent non-mathematician (or a mathematcian who is not an analytic number theorist) can understand and enjoy. Highly recommended.
Thanks Thud. I was about to post the very same thing.
This is coming from a secondary maths teacher trained as and engineer and having never had any formal tuition in number theory.