Re: Was Fermat\'s Theorem Really Proven?
 

As a previous poster mentioned, Wiles's proof has been around for over 10 years now and been presented to a lot more than a handful of mathematicians. When he first released it there were flaws to which attention was called almost immediately. Since the revised proof of 1994, no additional flaws or corrections have come to light in another 10 years of close scrutiny by thousands of individauls.

If you had asked this question in 1995, you could probably get odds of 100:1 for finding another flaw. Sometime in 1996 or early 1997 you could probably have gotten 100000:1 odds. But now, enough work has been done, enough alternative proofs given and extensions of Wiles's work made, that it's pretty much a done deal.

Frankly, I would rate the chance of a flaw in the proof four-colour theorem as higher than the chance of a flaw in Fermat. It was much less accessible than Fermat ever was. (But I won't give 100000:1 on it either.)

Something you might be interested in is the Great Internet Mersenne Prime Search's "proofs" of what the 31st to 38th Mersenne primes are, by either factoring or doing two Lucas-Lehmer tests with identical results for every candidate. The chance of a hardware failure causing a single test to come out wrong have been watched for several years and are on the order of 1 in 100; but the chances of two out of three tests (if the first two tests don't match, a third test is done, and whatever answer comes up twice is accepted as correct) on different machines both falling victim to hardware errors and giving the *same* wrong residue for the L-L test is down around 1 in 10^20. Millions of possible exponents have been tested now. So there is something like a 1 in 10^12 chance that there is one number with a wrong L-L residue in the GIMPS database, and something around a 1 in 10^7 chance that that number, if retested, would turn out to really be a Mersenne Prime. There is a mathematical proof you can bet on: how does 10^19:1 on "there are exactly 37 Mersenne primes smaller than 2^6972593-1" sound?

I am on a budget of course so I can't take toooo much action on this; but if you'll transfer me one attodollar (plus $3.95 shipping and handling) I will set aside ten bucks for you in case a smaller Mersenne prime ever turns up.

Re: Was Fermat\'s Theorem Really Proven?
 

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Personally I would take 100,000 to one odds that Wiles proof will eventually be shown to have a flaw. That's an exceedingly low number regarding a math proof, I think. But I am very unknowledgeable in this field. I wonder therefore what professional mathmeticans would make the odds.

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Personally, I believe the proof. I watched the Nova special about Andrew Wiles three or four times--I still have it. I didn't understand most of it, but it was fascinating none-the-less. Early on he thought he solved it, but there was a problem. That problem was solved. You should view this tape. You might be able to find it at your local library. Nova repeats their programs from time to time as well.

But here is a more interesting question. Fermat himself said he had simple proof of this theorem which the margins of his book was too small to contain.

Will anyone ever be able to come up with such a "simple" proof, and perhaps more importantly--was Fermat telling the truth?

Re: Was Fermat\'s Theorem Really Proven?
 

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Will anyone ever be able to come up with such a "simple" proof, and perhaps more importantly--was Fermat telling the truth?

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Proofs will get simpler over time. How much simpler is hard to say.

I think "almost everyone" agrees that Fermat thought he had a proof, but that it was, in fact, flawed just like the thousands of alleged proofs that came in the years after his book was published.

For that matter, a great many 17th and 18th century proofs, judged correct at the time, would now be considered flawed because of the increased rigor that came into fashion in the 19th century. For instance, if a student were to follow exactly the same steps that Euler used to "prove" that 1+1/4+1/9+..+1/n^2 = Pi^2/6, his paper would be marked wrong (for playing too fast and loose with factoring a polynomial of infinite degree) - Euler just didn't write down a justification for why he could get away with certain manipulations of infinite series, and now we find it easier to prove it by a different method that to fill in all of the background for why Euler's method worked.

Re: Was Fermat\'s Theorem Really Proven!
 

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Just an example of a mathematical theory that has been considered to be refuted is set theory. I don't claim to be a mathematical expert but I just thought I'd throw that out there.

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I'm pretty sure set theory is logically consistent. Are you referring to something like Russell's Paradox or the Continuum Hypothesis?

Re: Was Fermat\'s Theorem Really Proven!
 

I think they're talking about the fact that originally it was thought that set theory could encompass all of mathematics and make it 'complete'.

Re: Was Fermat\'s Theorem Really Proven?
 

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Will anyone ever be able to come up with such a "simple" proof, and perhaps more importantly--was Fermat telling the truth?

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Fermat later claimed less than Fermat's Last Theorem. If he really had a simple proof, he wouldn't have done that.

I don't know how simple a proof there will be at some point. I doubt there will be a something you can show a bright high school student in an hour. Something to keep in mind is that the statement of FLT fails in some extensions of the integers, so you need to use something that is true about the integers, and false for the extensions. Simple algebraic manipulations won't do that.

Re: Was Fermat\'s Theorem Really Proven?
 

I will lay you 1 quintillion:1 that there is no largest prime.

Edit: in all seriousness, Andrew Wiles' proof is taught in many universities as a graduate level seminar. So there are a lot more people that understand the proof as compared to when it was first published. If there was a flaw in it, we would have found it by now.

Re: Was Fermat\'s Theorem Really Proven?
 

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Personally I would take 100,000 to one odds that Wiles proof will eventually be shown to have a flaw.

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if you specify a time period, i will take your bet for 1$.

Re: Was Fermat\'s Theorem Really Proven?
 

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Unlike scientific theories that are sometimes eventually shown to be wrong, generally accepted math proofs have never, to my knowledge, been overturned. Some like the proof that there is no largest prime, or that the harmonic series converges are so obvious that the odds it isn't true are greater than one in a googol (but less than one in a googolplex). More complex accepted proofs might be a quintillion to one favorites to be true.

But Wiles proof is different. It is lenghty, complex, has no obvious connection to the original question, and most importantly has only been double checked by a large handful of people. Furthermore I believe there is a probability argument that would allow Fermat's Theorem to have no counterexamples because of "chance". Also its proof had escaped the best minds for 500 years.

Personally I would take 100,000 to one odds that Wiles proof will eventually be shown to have a flaw. That's an exceedingly low number regarding a math proof, I think. But I am very unknowledgeable in this field. I wonder therefore what professional mathmeticans would make the odds.

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I'm suspicious of Black-Scholes. It concludes that the value of an option is independent of any Trend in the stock, and as I see it does so by assuming an infinite bankroll that allows infinite hedging if the Trend goes against you. Since the Planet's entire options market is based on this Theorum you can actually get bets down if you want to gamble on a flaw.

PairTheBoard

Re: Was Fermat\'s Theorem Really Proven?
 

Many times mathematicians go about writing proofs to problems they feel is true. x^n + y^n = z^n seems true for every case you can think of.. the problem is proving the generality. If Wiles' proof is flawed.. it will be in how it's justified, and not because someone discovered a counter-example.

What's more interesting is proving something that mathematicians believed true for centuries isn't right... like when they tried to prove Euclidean geometry - then found out they couldn't.