Was Fermat's Theorem Really Proven?

[oneeye13]

what about his assertion that the search for odd perfect numbers is pointless? what odds can i get that that turns out to be wrong?

[jason_t]

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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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David,

I am a PhD student in mathematics. I sat through a semester long course on the proof of Fermat's theorem a couple of years ago. I set the odds that a flaw is shown to exist in the Taylor/Wiles proof at 10^100:1.

Historically, there are two examples of accepted math proofs being overturned that come to my mind; I am sure that there are others. In the last 1800s, a proof of the four-colour conjecture stood for approximately ten years and in the 1960s a proof of the so-called invariant subspace problem stood for a couple of years before it was overturned. A proof of the former was given by Appel/Haken in the late 1970s and the latter has still never been proven.

Also, I am sure it was just a typo, but the correct statement is that the harmonic series diverges, not converges as you wrote.

Jason.

[jason1990]

I'm a mathematician, but I'm not a number theorist, so some of what I'm about to say may be wrong. But here's my understanding. Wiles proved the Shimura-Taniyama-Weil conjecture. This was a deep conjecture in number theory, much bigger than Fermat's Last Theorem. The fact that STW implies FLT had been known for a long time. The proof that STW implies FLT is not incredibly long and complex. Wiles's original proof of STW did have flaws, which were found and corrected. Since then, a stronger version of STW has been proven by other people, which means there is yet another proof out there. I don't have the expertise to verify these proofs, but I personally know people who do and have. Given all of that, I think the proof of FLT is on very solid ground.

[BeerMoney]

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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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David, I'd take .00001:1 .... Do you see why??????????????

[mmbt0ne]

</font><blockquote><font class="small">En réponse à:</font><hr />
Historically, there are two examples of accepted math proofs being overturned that come to my mind; I am sure that there are others. In the last 1800s, a proof of the four-colour conjecture stood for approximately ten years

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Ok, If I remember correctly the four-color conjecture has been shown to be true through extensive computer modelling, but there isn't a proof per se to show it to be so.

[jason_t]

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Historically, there are two examples of accepted math proofs being overturned that come to my mind; I am sure that there are others. In the last 1800s, a proof of the four-colour conjecture stood for approximately ten years

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Ok, If I remember correctly the four-color conjecture has been shown to be true through extensive computer modelling, but there isn't a proof per se to show it to be so.

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The Appel/Haken proof reduced the problem to verifying the conjecture for roughly 1800 different cases which they programmed a computer to do. The computer program output that all 1800 cases were 4-colourable and thus the theorem was proven. A few years ago a similar proof was given that required roughly 500 cases to be checked.

[mmbt0ne]

Ok, here's a question I've never gotten a straight easy answer too. So, we know that if we have a graph with node in the shape of a wheel, it can't take more than 4 colors to make it so that no arc begins and ends on the same color.



So, why can't we look at a map as a collection of these wheel graphs, where the states/countries/whatever are the nodes, and the borders are the arcs?

[jason_t]

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Ok, here's a question I've never gotten a straight easy answer too. So, we know that if we have a graph with node in the shape of a wheel, it can't take more than 4 colors to make it so that no arc begins and ends on the same color.



So, why can't we look at a map as a collection of these wheel graphs, where the states/countries/whatever are the nodes, and the borders are the arcs?

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That is how the maps are viewed when thinking about this problem. Each country is represented by a node and adjacent countries have an arc drawn between them. The maps that satisfy the hypotheses of the four-colour problem turn out to be planar (none of the arcs cross) and the four-colour theorem is more commonly stated as any planar graph (that's what the object with the nodes and arcs is called) is four-colourable (you can colour the nodes with four colours so that no arc has the same colour at its two endpoints). Not all maps will necessarily reduce to such a collection of wheel graphs. Just draw an incredibly complicated planar graph and then it's easy to construct a map corresponding to that graph.

[pzhon]

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Just an example of a mathematical theory that has been considered to be refuted is set theory.

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As a mathematician, I have no idea what you mean when you say set theory has been refuted. I don't know of any problem with set theory. I suggest that you are remembering something that got garbled.

[gumpzilla]

Something that you might find interesting to read that's kind of related to this topic is Proofs and Refutations by Lakatos. The meat of the book consists of a dialogue between an imaginary teacher and students regarding "proofs" involving the Euler characteristic. I thought it was quite good.