Elegant Proof for Fermat's Last Theorem? (1 Viewer)

[hyparzero]

Fermat's last theorem was proved by Wiles by first prooving the shimura-taniyama conjecture, however, the final proof was more than 150pages, and it was claimed Fermat had a much more elegant ptoof to his theorem, which stated there are no positive integers which satisfy the following condition if n>2

xⁿ + yⁿ = zⁿ

If one has such an elegant proof, how does one submit it for evaluation.

 

[Dumsum]

By posting it here. Maths nerds all around the world patrol this forum. True story.

 

[hyparzero]

and let everyone else steal it? I mean mathematical instituitions.

 

[Templar]

You can post or submit it to any mathematical institute, such as an university. However, that's largely irrelevant as I doubt you will come up with a much simpler proof to the scale of the one Fermat possibly used.

Fermat's Last Theorem was proved using a specialised case of Shimura Taniyama conjecture. The entire conjecture was proved later by Taylor et al, with Taylor the one who helped Wiles patch up his proof.

 

[hyparzero]

Hmmm... i played with the theorem a bit, and what i've done is:

I took on the assumption that the solutions x,y,z to be integers > 0

Then I proved that the ratio of (x,y,z) to one another are always irrational for n>2

Thus, if the ratios are irrational, then (x,y,z) cannot be integers, and hence, there are no integer solutions to the condition:

xⁿ + yⁿ = zⁿ

which proves Fermat's Last Theorem.

Does anyone see a flaw with my reasoning without me going into the details of the proof.

 

[SeDaTeD]

If you have indeed proved that the ratios are irrational, then it seems fair enough to me. If it is true, I'd be surprised if nobody has ever come up with your method before. Try to see if you've missed anything or if there are any flaws. If you are sure there are none, then congratulations .

 

[STx]

heres one i found: http://www.economics.ox.ac.uk/Members/giuseppe.mazzarino/Fermat_March_2003.pdf

 

[Templar]

Partial cases of Fermat's Last Theorem has been proved by reductio ad absurdum, but it's highly unlikely the entirety could be proved as such.

 

[buchanan]

Partial cases of Fermat's Last Theorem has been proved by reductio ad absurdum, but it's highly unlikely the entirety could be proved as such.

Actually, Wiles' proof is a reductio ad absurdum (albeit a very abstract one):

All Frey curves are nonmodular and all semistable elliptic curves over Q are modular. (In fact all elliptic curves over Q are modular.) A Frey curve exists if an only if Fermat’s Last Theorem is false. If Frey curves exist, they are nonmodular semistable elliptic curves over Q. But all such curves are modular - contradiction. Therefore Frey curves do not exist and therefore Fermat’s Last Theorem is true!

<a href="http://users.tpg.com.au/nanahcub/summary.pdf">See my summary for more details</a>

<a href="http://users.tpg.com.au/nanahcub/flt.pdf">- and Wiles' complete proof</a>

<a href="http://math.stanford.edu/~lekheng/flt/bcdt.pdf">- and the full proof of the Taniyama-Shimura-Weil conjecture</a>

 

[Templar]

What I meant was assuming that x, y and z are integers and then proving there will always be a smaller set, as opposed to the induction used to prove modular curves.

 

[hello2004]

I think I found a very large error in your solution:

Then I proved that the ratio of (x,y,z) to one another are always irrational for n>2

Consider the equation:

x^3 + y^3 = z^3

and let a particular solution be:

x = squareroot(2)
y = squareroot(2)
z = 2^(5/6)

... the ratio of x to y is '1' and is obviously not irrational. I think your solution is invalid as x and y can be linearly related by an infinite amount of constants to which a value 'z' will exist.

 

[hyparzero]

I think I found a very large error in your solution:



Consider the equation:

x^3 + y^3 = z^3

and let a particular solution be:

x = squareroot(2)
y = squareroot(2)
z = 2^(5/6)

... the ratio of x to y is '1' and is obviously not irrational. I think your solution is invalid as x and y can be linearly related by an infinite amount of constants to which a value 'z' will exist.

But the ratio of x to z clearly is.
I proved that all you needed is one ratio to be irrational for all solutions (x,y,z) to be irrational.

The original theorem states no integer solutions exists for (x,y,z), such that for any irrational solution of either x,y or z, the theorem is deemed true.

 

[buchanan]

There's an easy way and a hard way to learn about the proof of Fermat's Last Theorem.

THE EASY WAY

The easy way is just to watch the videos and read popular accounts.

For many years, msri have had a video online

Google video has recently added the uktv video to the internet which is better than the msri one

Here they are: <a href="http://video.google.com.au/videoplay?docid=8269328330690408516">uktv</a> ; <a href="http://ia300112.us.archive.org/2/items/fermats_last_theorem/fermats_last_theorem_256kb.mp4">msri</a>

If you want a higher quality version of the uktv one, several torrents of it are available online.

Note that the uktv (bbc/wgbh) one has Eve Matheson narrating and is slightly out-of-sync. There also exists an identical nova version (pbs) narrated by Stacy Keach instead, called "The Proof", but thus far I don't think the nova one is online. This one is better (in-sync). I've got both versions. Although the nova version isn't online, nevertheless the transcript for it is: http://www.pbs.org/wgbh/nova/transcripts/2414proof.html and nova has made a whole website for it at http://www.pbs.org/wgbh/nova/proof/

Even though the uktv one is better than the msri one, I still think it is good to watch them both.

There are other videos from the ams which are not online yet and are also good to watch, but are more technical:

"Modular Elliptic Curves and Fermat's Last Theorem." Kenneth Ribet; August 1993. American Mathematical Society, 1993. (Available at http://www.ams.org/bookstore?fn=20&arg1=videos&item=DVD-89)

"Fermat's Last Theorem." Barry Mazur; AMS-CMS-MAA Joint Invited Address, August 1993. American Mathematical Society, 1995. (Available at http://www.ams.org/bookstore?fn=20&arg1=videos&item=VIDEO-97)

There also is a more theatrical video called Fermat's Last Tango available at http://www.claymath.org/publications/Fermats_Last_Tango

Popular accounts:

Singh, S., Fermat's Last Theorem, Fourth Estate, 2002

Azcel, A. D., Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem, Thunder's Mouth Press, 2007

Ribenboim, P., Fermat's Last Theorem for Amateurs, Springer, 1999 (note there is a 2nd edition in 2000)

Mozzochi, C. J., The Fermat Diary, AMS, 2000

Mozzochi, C. J., The Fermat Proof, Trafford Publishing, 2006

THE HARD WAY

The easy way raises awareness with the public but as Andrew Wiles himself has said there is simply no substitute for hard work. And the hard way is as follows:

If you only want the mathematics needed to cover Wiles' proof, then you should read his proof (which I mentioned in <a href="http://community.boredofstudies.org/2436198/post-9.html">this post</a>) and references contained therein. Also, since then a textbook has been written to facilitate this process. It was originally written in French, but was soon translated into English:

Hellegouarch, Y., Invitation to the Mathematics of Fermat-Wiles, Academic Press, 2001 (English translation by Leila Schneps, 2002)

Unlike most books on the subject this one contains a large number of exercises to assist you to not only read it, but more importantly, to actually learn it.

Nevertheless you will need to read more widely if you want a firmer grasp of the mathematics used in Wiles' proof.

A good start are the references Ribet mentions at the end of his 1993 video but does not give details for. Here are those details:

Lang, S. and Rubin, K., Cyclotomic Fields I-II, Springer, 1989.

Silverman, J. H. and Tate, J. T., Rational Points on Elliptic Curves, Springer, 1992 (note also that there is 2nd edition in 1994)

Silverman, J. H., The Arithmetic of Elliptic Curves, Springer, 1986 (note there is a 2nd edition in 1994)

Cornell, G. and Silverman, J. H., Arithmetic Geometry, Springer-Verlag, 1986 (note there is a 2nd edition in 1998)

Knapp, A. W., Elliptic Curves, Princeton University Press, 1992.

Hearst III, W. R. and Ribet, K. A., Book Review for Rational Points on Elliptic Curves by Joseph H. Silverman and John T. Tate., AMS Bull., Vol. 30 No. 2, April 1994, pp 248-252.

In 1995 a conference was held on August 9-18 at Boston University to cover all the maths Wiles used and the notes were published in

Cornell, G., Silverman, J. H., Stevens, G, Modular Forms and Fermat's Last Theorem, Springer, 1997 (note there is a 3rd edition in 2000)

This is the definitive book on the issue. But it's best to consult the other references I've mentioned first.

 

[buchanan]

Here's a more recent video by Darmon (2006):
http://www.uni-math.gwdg.de/aufzeic...vi/SummerSchool_20060728-1600_Darmon_xvid.avi (545 Mb)

 

[buchanan]

<b>Correction to Proof of Fermat's Last Theorem.</b>

In line 7 on page 495 on older versions of the online pdf file, <img src="http://users.tpg.com.au/nanahcub/incorrect.gif" /> should be replaced with <img src="http://users.tpg.com.au/nanahcub/correct.gif" />

The file <a href="http://users.tpg.com.au/nanahcub/flt.pdf">http://users.tpg.com.au/nanahcub/flt.pdf</a> has now been corrected.

It is glaringly obvious, but for some reason has eluded everyone for the 15 years it's been on the net till now!

I thank Andrew Morrow for bringing this to my attention.