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Hi Jason,

I have just finished 'Fermat's Last Theorum' by Simon Singh and was just wondering if I could get your opinion as a pro mathematician.

It seemed to me (as a layman) that since Professor Wiles' proof used modern mathematics not available at the time of Fermat, that Fermat almost certainly couldn't have had a proof of it. If it required a new branch of mathematics to prove, then I would imagine that there is no simpler solution available. It gives an example of Fermat proving a specific case (I think n=4, but can't remember for sure). Do you think that Fermat thought this was a general solution?

What I am trying to ask is, in your opinion, could Fermat really have had a proof for it? If so, why has nobody been able to find a proof that would have been possible in Fermat's time?

Thanks

No, of course not. I'm not a number theorist but I'm right about this.

Fermat claimed to have a proof (see below) and if he did, [Did he have a reason to lie about this?]; I submit that his proof would have been much simplier than the submitted modern solution. Whether this hypothesis is true or not time will tell. Perhaps in 100 more years someone will come up with a proof that will only take one page of foolscap.


Link to Fremat's Last Theorem:

http://mathworld.wolfram.com/FermatsLastTheorem.html

From the above link:

"Fermat's last theorem is a theorem first proposed by Fermat in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus. The scribbled note was discovered posthumously, and the original is now lost. However, a copy was preserved in a book published by Fermat's son. In the note, Fermat claimed to have discovered a proof that the Diophantine equation X^n + Y^n = Z ^n has no integer solutions for n>2 and X,Y,Z not equal to Zero.


The full text of Fermat's statement, written in Latin, reads "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet" (Nagell 1951, p. 252). In translation, "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain." '


-Zeno

PS - Also from above link:

"The proof of Fermat's Last Theorem marks the end of a mathematical era. Since virtually all of the tools which were eventually brought to bear on the problem had yet to be invented in the time of Fermat, it is interesting to speculate about whether he actually was in possession of an elementary proof of the theorem. Judging by the tenacity with which the problem resisted attack for so long, Fermat's alleged proof seems likely to have been illusionary. This conclusion is further supported by the fact that Fermat searched for proofs for the cases and , which would have been superfluous had he actually been in possession of a general proof. "

**

So my above hypothesis seems to have less weight and must pass down to just a conjecture. It may still be possible that Fermat had a general proof, but seems less likely than I purposed.

Its pretty much considered a fact that Fermat was wrong about the proof.

My father is a probabilist. Can you come up with a clever birthday gift for him?

Give him one of two envelopes, one containing twice the amount of money as the other, and ask him if he wants to swap.

haha this is pretty good

Re whether Fermat had a proof...

If we mean something that Fermat's contemporaries would have accepted as a proof, I think it remains possible-but-unlikely. If we mean something that would be acceptable still today, almost surely not. Most of the "proofs" through Euler's time and even into the early19th century wound up requiring some additional gaps plugged in them when the new post-Cauchy standards of rigor became accepted.

There does still remain a chance, even a likelihood, of a much simpler proof being found in the future. One page? Maybe not. But there are a LOT of problems that were first solved by a tortuous path and later a more elegant way was found.

The following question was sent to me by PM, and is posted here with permission. I will try to give a reply in the near future. In the meantime, if anyone wishes to comment on it, please do.

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You have 500 unique units. You select 30 at a time, and you do this a total of 4 times. At the end, you could have selected a maximum of 120 unique units, and a minimum of 30 units (e.g. you pick the same 30 units each time).

Assuming the selection process is a simple random sample without replacement for each group of 30, after four such selections (with replacement in between selections), what are the probabilities of observing...

For X, Y, & Z in {0,1,2,...,30},

X units 2 or more times?
X units 3 or more times?
X units 4 times?
X units 2 times, and Y units 3 or more times?
X units 2 times, Y units 3 times, and Z units 4 times?

Note: A very simple start on this problem is of course to examine 2 selections instead of 4. And moving up one notch in complexity, examining the probabilities of observing the same unit 3 times over 3 selections is also pretty straight forward to calculate...

For added difficulty, what if there are, say, 500+N unique units, and a (non-random) possibly different subset of units are available for selection each time? For example, for the first selection, units # 400-414 are not available; for the second selection, units # 401-410, and 511-515 are not available, etc. And what if the number selected each time varies? For example, the first time 31 are selected, the second time 26 are selected, etc.

Yes, I believe it is possible. I believe it is quite unlikely, though. Most likely, he thought he had a truly marvelous proof and was very proud of himself, but in fact the proof did not work. I think most mathematicians know what that feels like.

It is actually pretty common for theorems to be reproved through different means. The proofs we read in textbooks are often much simpler and shorter than the original proofs that one would find in the literature. This is probably more common with "major" theorems. Major theorems are constantly in use. People are applying their results in many different fields, trying to generalize them or modify them for different settings. This makes it more likely that someone will come up with a new and better proof.

I do not foresee this happening with FLT, though. In other words, even if there exists a proof which uses only mathematics available in Fermat's time, I doubt we will ever find it.

not a probability question but i saw the few posts on fermat and i am curious. taking a look at wiles' 100+ page proof, i don't understand a single page. to me, it's nearly 100% gibberish. as a PhD mathematician, even though number theory or whatever isn't your field, how much of the proof do you understand?

if it's essentially none, do you consider yourself "stupid" for not being able to understand it?
if it's a decent amount, how long do you think it would take you to be able to fully grasp the proof?
if it's most or all, where did you attend undergrad, and just how easy were your math courses (to you)? basically, when do you start learning the material needed to understand the proof? i just graduated from a top university (though not a mathy school) very near the top of my class, and i can't see a single person being able to understand any of the proof. i'm having a hard time grasping how the proof for an easily understood problem can be ENTIRELY INCOMPREHENSIBLE to otherwise intelligent people.

I am really bad at this sort of thing. Actually, for Father's Day, I hope I get something that has nothing to do with probability or mathematics!

I like lastcardcharlie's suggestion a lot! You could get two identical greeting cards and write a nice note. That sounds really cool, actually.

Another possibility is some kind of popular book about probability, preferably one that is surrounded by some kind of controversy. For me, I am somewhat curious what that Black Swan book has to say (even if I end up thinking it is nonsense), but I would probably never spend money on it for myself. I know a certain probabilist recently published some scathing critiques of Dembski's No Free Lunch book on intelligent design.

For Christmas, someone gave me the math mug, although my first impression was that it looks more like a physics mug. You could have a mug like this custom made, and put on it many important formulas from probability theory.

There is a pretty good chance we know what his incorrect proof was based on, http://en.wikipedia.org/wiki/Fermat%...neral_proof.3F. I don't think it is very likely at all that he had an ok proof that just wasn't completely rigorous, I think its much more likely he was completely wrong.

1 page and using only math known at the time of Fermat seems almost impossible. There could always be new math discovered in the future that makes a difficult proof trivial.

Thanks for the info guys.

Here is a partial answer, which may get you moving in the right direction. For 1 ≤ n ≤ 500, let X_n be the number of times unit n was selected. The number of unique selections is

Here 1_A is the indicator function of the event A. It is the random variable which is 1 if A occurs, and 0 otherwise. According to the above formula, the expected number of unique selections is

The probability that the first unit is selected in the first round, and then not in any other round is (30/500)(470/500)³. Hence, P(X₁ = 1) is 4 times that. This means the expected number of unique selections is

500(4)(30/500)(470/500)³ = 99.67008.

The expected number of duplicates, triplicates, and quadruplicates can be analyzed in a similar fashion. If you want a more detailed analysis that goes beyond expectations, that will be more difficult as you will then have to delve into the correlations between the above indicator functions. If you really want to go down that road, then inclusion-exclusion may be very useful to you. But perhaps simulation would be easier in the end, depending on what you want to get out of this analysis.

I understand enough of the proof that I could give the Cliffs Notes version to a graduate student. You would probably not begin to learn the necessary prerequisites until graduate school, and even then, only if you intended to pursue that area of research. Modern mathematics is an enormous discipline and modern mathematicians are highly specialized.

You may find this article interesting: Proof and Beauty. A quote from the end of the article:

So do you understand the formulation of the Birch and Swinnerton-Dyer Conjecture?

http://www.claymath.org/millennium/B...er_Conjecture/

Cliffs Notes?

Jason, why '1990'?

He's the worlds youngest professor.

Funny. I'd like to think of myself as a pretty competent 18 year old but had Jason been my age I would've been thoroughly humbled.

This I am not familiar with, but from some superficial looking around, I am guessing that the Shimura-Taniyama–Weil conjecture is a special case of it. The article cited below gives a good Cliffs Notes introduction to Shimura-Taniyama–Weil.

"A Marvelous Proof"
Fernando Q. Gouvea
The American Mathematical Monthly, Vol. 101, No. 3 (Mar., 1994), pp. 203-222
(article consists of 20 pages)

Stable URL: http://www.jstor.org/stable/2975598

Asked and answered.

Thanks. I accessed it through my library.

For anyone else who might be interested in this article, here is a free copy I just found: http://math.stanford.edu/~lekheng/flt/gouvea.pdf