Has it been proved or disproved? Some mathematicians claim they proved it for whole intergers, n, such that n>2. Another bevy of mathematicians claimed they disproved it. One enthuasist, in particular, claimed he found series of numbers for powers higher than 2, but with a 0.01% idiosyncrancy, which means that the values he picked were approximations, and that doesn’t really prove anything-there’s a potential for discrepancies in the math. Who’s correct?
You’re a bit behind the times.
Andrew Wiles came up with a proof in 1993, which, with a bit of tinkering, was soon accepted.
There are several books out on the subject by now.
One can’t help but wonder if a more elegant solution than Wiles’ still awaits discovery.
Maybe, maybe not. He may have found the shortest proof.
Actually I have found a simpler, more elegant proof, but I just can’t fit it into the margins of this posting. 
Some people think old Fermat really had a proof up his sleeve, but he probably had one of the many wrong proofs that have been offered up over the centures. The math of his time wasn’t capable of producing the proof Wiles and company created, so it’s pretty reasonable to think that Fermat couldn’t have had anything other than a false hope.
Of course, mathematicians can be as hopeful and as glory-seeking as the rest of us, so I’m betting someone is still looking for a proof in math Fermat could have used. Just because it’s been proven doesn’t mean the book is closed on the subject.
And Fermat almost certainly knew this. What many people don’t realize is that Fermat lived for another twenty eight years after jotting down his famous note. If he had really discovered a short proof, or even if he had mistakenly thought he had, he had ample opportunity to publish it. What probably happened was that Fermat had an idea, jotted down the note, realized the next morning there was a flaw in his proof, and then forgot about the note he had written.
The lure of the theorem is that it hinges on a rather remarkable mathematical fact, which most people can grasp (I use an Excel worksheet to explain it), and it just seems like the proof is teneble. Instead, it’s really mind-numbingly complicated and incomprehensible to most people (including me). I love the idea of someone – maybe a child prodigy, acting on pure instinct – will come up with a one-page explanation of why there’s no Pythagoras-type whole number sets for any exponential series after two.
The flip side of this is that a low-machinery proof is not often the best proof.
I just proved the Fundamental Theorem of Algebra (every polynomial over C has a root) to my multivariable calculus students as an application of Green’s Theorem, partly because it can be done with just the machinery a third-semester calculus student has. However, the most enlightening proof (which shows what’s “really going on” uses the machinery of deRham cohomology on C and isn’t even presented to most undergrads who take a course in complex analysis.
This is to say: There may be a proof of Fermat’s “Last Theorem” which is graspable by, say, a bright undergraduate student, but I’ll take the T-S-W route as more important any day.
Actually, I don’t think the fact that he never published a solution (even though he had ample time) is necessarily a good argument that Fermat realized his solution had a flaw. I’m not sure of Fermat’s time in particular, but the academic world has not always been focused on publishing. In past times, it was not uncommon for researchers to actually hide their results–one could make a reputation for himself with the ability to solve problems no one else could. The story of Tartaglia, Cardano, and the solution of the cubic is a perfect example of this (from the 16th century, versus Fermat in the 17th century, so I would tend to expect this attitude still existed in Fermat’s time):
They can also be mischievious, and I’ve always suspected this was his idea of a joke to provoke exactly the kind of speculation we see here. IOW, a theoretical troll.
Hi. Could you post the url for this proof?
There is an algorithm for solving cubics, which I learned in high school-it involves eliminating the second term of the equation, and making a number of substitions, and solving quatradics.
Are there any other mathematical conjectures worth scrutinizing? Does anyone know the algorithms for solving quartic equations?
[Senator Claghorn voice]It’s a joke, son.[/voice]
There is also the fact that, after writing his infamous margin note, he went on to prove the theorem for the special cases of n=3 and n=4 (if I remember right). He presumably wouldn’t have bothered with the special cases if he already had a proof for the general case.
But I suppose he might have, just for the mental exercise.
Dr Starfish
I guess you are not familiar with the famous story about Fermat saying he had the proof but couldn’t fit it into the margin of the page on which he was writing.
Some people who have posted here think Fermat had the proof but found it was faulty. I have heard that he never had the proof at all (flawed or otherwise).
I have also read that Fermat was a talented amateur in mathematics. He had brilliant insight into setting up conjectures, theorems which were extremely difficult to prove or disprove. Fermat claimed to have developed a formula that would always generate prime numbers. (Look up Fermat primes for more information). The 5th prime from that formula is 65,537 and the sixth was so big, nobody could prove or disprove its primality. About a century after Fermat’s death, Euler determined that the 6th Fermat “prime” was NOT prime at all. But of course Fermat was long gone by then and so his “prime formula” was regarded as valid in his lifetime. His ability to postulate brilliant conjectures that required frustratingly difficult proofs prompted one of his comtemporaries to call him “that French bastard !!!”
You may be right about the general practice, but Fermat published numerous proofs in his lifetime. There’s no reason to suppose he wouldn’t have published this one as well.