# Fermat's Last Theorem



## MW1990 (Aug 23, 2009)

Hello everyone  I first watched a documentary on this glorious topic in high school, and I wanted to share it with you  This documentary, Andrew Wiles, and Terry Tao, along with many other people and topics inspired me to pursue studies in the mathematics field.

In this documentary, Andrew Wiles proves the Taniyama-Shimura conjecture, in which all elliptic curves are modular, thus proving Fermat's Theorem that there are no nontrivial solutions to the diophantine equation:

x^n + y^n = z^n, 
where (x,y,z) cannot equal zero (trivial solution) and 
where n > 2 (excluding some special cases that have been proven by Germain and others).

Enjoy!!

http://video.google.com/videoplay?docid=8269328330690408516


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## calekewbs (Aug 23, 2009)

im 23 minutes into this. this is pretty cool! lol


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## MW1990 (Aug 23, 2009)

yep  Pure mathematics is very interesting


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## calekewbs (Aug 23, 2009)

yeah it is. lol have you looked at my thread about a hard riddle? if not, go check it out. I think you would be interested.


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## MW1990 (Aug 23, 2009)

Oh the Scheherazade one?  If so, I am interested!


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## nitrocan (Aug 23, 2009)

I know that two mathematicians were working about this for 6 months and then presented their proof to someone (apparently there's a jury or something like that for this). After they corrected some trivial mistakes, they finally proved it.
It took like 2-3 days or something to explain it all.


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## Cride5 (Aug 25, 2009)

nitrocan said:


> I know that two mathematicians were working about this for 6 months and then presented their proof to someone (apparently there's a jury or something like that for this). After they corrected some trivial mistakes, they finally proved it.
> It took like 2-3 days or something to explain it all.



It was Andrew Wiles who had the final breakthrough. In the documentary he said he had been actively working on the problem for 7 years! I think he brought his colleague Richard Taylor on board after the original (flawed) proof was announced, to help fix it. The main body of work (and credit) certainly belongs to Andrew Wiles.

An interesting question which still hangs though, is how Fermat was able to prove it using 17th Century mathematics?!



nitrocan said:


> I know that two mathematicians were working about this for 6 months and then presented their proof to someone (apparently there's a jury or something like that for this). After they corrected some trivial mistakes, they finally proved it.
> It took like 2-3 days or something to explain it all.


My complex variables prof was at the colloquium where Andrew Wiles first presented his proof publicly. Wiles' presentation spanned two full days, but was titled something really cryptic, so only a dozen or so serious pure math geeks were at the lecture. By lunch time a couple of people started to suspect where the proof was going and the rumors flew until by the end of the first day everyone was calling their colleagues and people were jumping on airplanes to come to the second half. The next day was completely packed, standing room only and Wiles got a standing ovation at the end of it.

It turned out there were some problems with the proof, but he went back and fixed them over the course of a few months, but was finally triumphant. It was a pretty cool story, if you're a math geek.


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## brunson (Aug 25, 2009)

If you're interested in the history behind all the math that had to be invented before Fermat could be proven, this: http://www.amazon.com/Fermats-Last-Theorem-Unlocking-Mathematical/dp/1568583605/ is an awesome, awesome book.

Basically, the author explains how *much* of the math that went into the proof didn't exist when Fermat proposed his theorem. It would be a truly amazing proof if Fermat could have fit it into the margin of a notebook like he claims to have done. It's generally accepted that he must have later realized his proof was incorrect and never brought it up again.


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## Johannes91 (Aug 26, 2009)

brunson said:


> It would be a truly amazing proof if Fermat *could have fit it into the margin* of a notebook like he claims to have done.


It was exactly the opposite.

_"I have discovered a truly marvellous proof of this, which *this margin is too narrow to contain*."_


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## mazei (Aug 26, 2009)

Well looking at what Andrew Wiles had to do, I think the proof wouldn't fit in the margin at all. Perhaps Fermat wrote the proof somewhere else but lost it.


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## blah (Aug 26, 2009)

mazei said:


> Well looking at what Andrew Wiles had to do, I think the proof wouldn't fit in the margin at all. Perhaps Fermat wrote the proof somewhere else but lost it.


:fp


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## qqwref (Aug 26, 2009)

Given that the FLT proof required extremely advanced mathematics that wasn't even around 50 years ago, I doubt Fermat had a proof at all. As brunson says it's more likely that he simply thought he had a simple proof which ended up not covering all cases, as the problem was much more complicated than he could have anticipated. (This sounds unlikely but has happened before - the four-color theorem was "proved" in 1879 and 1880, using two different techniques, but both arguments were shown to be incorrect about a decade later, and it took about 100 years for a real proof to come along.)


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## Johannes91 (Aug 26, 2009)

What makes all of you think that there can't exist a proof that's simpler than Andrew Wiles'?


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## deepSubDiver (Aug 26, 2009)

you pointed it out, johannes. there might have even been such simple general proof theory that got lost during decades, which fermat applied for that proof and is not known today.


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## mazei (Aug 26, 2009)

blah said:


> mazei said:
> 
> 
> > Well looking at what Andrew Wiles had to do, I think the proof wouldn't fit in the margin at all. Perhaps Fermat wrote the proof somewhere else but lost it.
> ...



Cmon, you really think I actually believe that? Just a joke bro.


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## brunson (Aug 26, 2009)

Johannes91 said:


> What makes all of you think that there can't exist a proof that's simpler than Andrew Wiles'?


I misremembered the quote, so you're right, he didn't claim that it fit in the margin. However, your straw man aside, the crux of my point was "It would be truly amazing if Fermat had a valid proof of the theorem at the time." There's nothing to say he couldn't have found a more elegant proof, it would be cool to see. Yet, why he didn't bother writing it down sometime in the next 28 years makes it seem as though it may not have existed.


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## Kian (Aug 26, 2009)

brunson said:


> Johannes91 said:
> 
> 
> > What makes all of you think that there can't exist a proof that's simpler than Andrew Wiles'?
> ...



That and the fact that it took 358 years for anyone to develop a sound proof, despite countless efforts, using math Fermat couldn't have dreamed out, makes me think it's extraordinarily unlikely Fermat had a proof at all.


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## qqwref (Aug 28, 2009)

Johannes91 said:


> What makes all of you think that there can't exist a proof that's simpler than Andrew Wiles'?



Since you're asking - the hundreds of brilliant mathematicians over the centuries who put significant time and effort into trying to discover a proof since the conjecture was published. If there was a simpler proof using any or all of the mathematical techniques that had ever been conceived of when Fermat died, I think it's not unreasonable to assert that someone would've thought of it, or at least come close.


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## Johannes91 (Aug 28, 2009)

qqwref said:


> Johannes91 said:
> 
> 
> > What makes all of you think that there can't exist a proof that's simpler than Andrew Wiles'?
> ...


Thanks for answering. I was just wondering whether there is a deeper reason than that.


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