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Q: Fermat's Theorem ( Answered,   8 Comments )
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 Subject: Fermat's Theorem Category: Science > Math Asked by: damoruso-ga List Price: \$4.00 Posted: 22 Apr 2002 08:58 PDT Expires: 29 Apr 2002 08:58 PDT Question ID: 2636
 ```Could the current solution to Fermat's Last theorem be the solution that Fermat derived? If not, why?```
 Subject: Re: Fermat's Theorem Answered By: drdavid-ga on 22 Apr 2002 16:57 PDT
 ```The Comments posted below summarize much of the prevailing view on your question. A good detailed history (for the mathematically literate!) of Fermat's Last Theorem can be found in the article "Fermat's Last Theorem," by J J O'Connor and E F Robertson, February 1996, which can be found on-line at http://www-groups.dcs.st- and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html They note that "it may well be that Fermat realised that his remarkable proof was wrong, however, since all his other theorems were stated and restated in challenge problems that Fermat sent to other mathematicians. Although the special cases of n = 3 and n = 4 were issued as challenges (and Fermat did know how to prove these) the general theorem was never mentioned again by Fermat." The article goes on to describe the subsequent history (Fermat's famous marginal note is thought to have been written around 1630) through the solution of Wiles which he worked on intermittently from when he first became fascinated with the theorem at the age of 10 in 1963. He was finally awarded the Wolfskehl Prize (unclaimed for almost 90 years) in 1997 for his solution. See the European Digest, Vol. 1, February-March 1998 for a discussion of the prize: http://www.european-digest.com/ecd01/docs/digest18.htm While it is difficult to be definitive--no one knows for sure what Fermat did or didn't know, there seems to be near-universal agreement that 1) Fermat probably did not really have a complete solution to his theorem, 2) The solution of Wiles is very unlikely to have been very close to whatever ideas Fermat did have. Many additional sites discuss Fermat's Last Theorem. These can be found simply by doing a Google search on: "Fermat's Last Theorem"```
 Subject: Re: Fermat's Theorem From: mgriffiths-ga on 22 Apr 2002 09:13 PDT
 ```No. The current solution runs to hundreds of pages, whereas Fermat referred to a simple and elegant solution. For example, see http://www.european- digest.com/ecd01/docs/digest18.htm "Wiles' solution, though, is not Fermat's solution. Though Fermat had said the margin of the book was too small to contain his remarkable proof, he did write to a colleague that he had devised a relatively simple way to solve it. Wiles' solution, on the other hand, runs to over 100 pages, is complicated and uses techniques not known in the 17th century."```
 Subject: Re: Fermat's Theorem From: josef-ga on 22 Apr 2002 11:50 PDT
 ```It is also possible that the solution Fermat had in mind might have seemed correct to him at the time, but it may have contained errors that would have been exposed if he had released it to the public.```
 Subject: Re: Fermat's Theorem From: scott_sauyet-ga on 22 Apr 2002 12:49 PDT
 ```And, given the copious research done on this problem, the latter seems much more likely.```
 Subject: Re: Fermat's Theorem From: johnfrench-ga on 22 Apr 2002 13:03 PDT
 ```Remember that we have only Fermat's comments to support the claim that he did indeed solve the problem. Although it is extremely unlikely that the current solution is the same one that Fermat conceived, it is impossible to prove that it is not. Remember, also, that we have no knowledge of his ever actually deriving it (putting all the steps to paper), let alone having conceived it.```
 Subject: Re: Fermat's Theorem From: mr_sheel-ga on 31 May 2002 08:21 PDT
 ```It's impossible for the current solution to have been Fermat's solution. Wiles uses a lot of "Galois Representation Theory" and ties together two areas of matheamtics that were only conjectured to be similar. All stuff that has been developed in the last 20 years or so. Fermat could not have done that because he was missing about 300 years of mathematics development.```
 Subject: Re: Fermat's Theorem From: alpertron-ga on 14 Jun 2002 08:40 PDT
 ```While the case n=4 was finally proved by Fermat himself, he was not able to solve the problem for other exponents. The case n=3 was solved by Euler. It was not uncommon for Fermat to write conjectures as if they were theorems. For instance, he stated in 1640 that the expression 2^(2^n)+1 is prime for every n>0. But this is obviously incorrect, because 2^(2^5)+1 = 4294967297 = 641 x 6700417. These numbers are now known as Fermat numbers and you can see their known factors at: http://www.prothsearch.net/fermat.html```
 Subject: Re: Fermat's Theorem From: gary8smith-ga on 20 Jun 2002 12:13 PDT
 `Can't someone just feed the problem in to a computer, and let it do the working out?`
 Subject: Re: Fermat's Theorem From: zhiwenchong-ga on 26 Jun 2002 22:05 PDT
 ```>Can't someone just feed the problem in to a computer, and >let it do the working out? Nope. Computers can't find proofs. If you mean checking the FLT for a single case where it breaks down, well, people have tested the equation it up to large large numbers without success... Knowing full well that the upper bound is infinity, it was an exercise in futility.```