Fermat's Last theorem

[ron_jay]

We all know of this theorem which was finally proved in the 1960's. It says that we cannot find any real integral solution for n>2 when an integer is expressed to a power of 'n' and is equal to the sum of two numbers which individually are raised to the power 'n'.

x^n=a^n+b^n

Well for n=2, we are familiar with the pythagorean(3,4,5 etc.) combinations, but there is indeed is no solution when n>2...check it out.

I recently ran a program to find this solution, but couldn't find for n=3,4,5... This certainly validates the theorem but how do we prove it mathematically?

[cristo]

I thought it was proved in the '90s?

[mgb_phys]

Either try all infinite 'n' - might take a while (actually only need to test all the infinite number of prime n's)
or learn an awful lot of seriously complicated number theory about elliptic curves and some conjecture that I can't even spell as Andrew Wiles did. There isn't a proof understandable by mere mortals.

[ron_jay]

Yes, thats a mistake, it was done so in the 90's but that's apart from the point...

[mgb_phys]

There is a good book by Simon Singh, but since it is pretty impossible to even outline the numerical techniques to someone without grad school maths it mainly concentrates on the stories of the people involved.

It would be interesting to know what Fermat's original proof was and where he went wrong!

[HallsofIvy]

There was no original proof. After he wrote his comment in the margin of a book, he gave completely different proofs for n= 3 and n= 4. He wouldn't have done that if he had a proof for all n.

What happened to Fermat is what happens to mathematicians all the time- he saw a way of extending result he already had and wrote that "marginal" comment. Later he realized it didn't extend as he thought.

[genneth]

Also known as the Taniyama–Shimura conjecture. Don't even bother starting unless you already know what elliptic curves and modular functions have to do with each other. (I don't. Not a clue.)

[HallsofIvy]

I wouldn't say it is "also known as". Yes, Wiles proved the Taniyama-Shimura conjecture. It has already been proven that Fermat's last theorem was true if and only if theTaniyama-Shimura conjecture was true.

[Kummer]

The German mathematician Frey announced that he believes that Fermat's equation, if false, will imply that the Taniyami-Shimura Conjeture to be false. This became known as the Eplison Conjecture. Soon the mathematician Kenneth Ribet actually prove this idea in detail. Wiles immediately realized that all he needs to do know if prove the Taniyami-Shimura Conjecture.

In the old days, Fermat's problem was attacked by Kummer Ideal Complex numbers. But this does not work succesfully. There is a book (I forgot the name) which teaches algebraic number theory while simultaneously working with Fermat's equation in a classical approach.

[Dragonfall]

I recently ran a program to find this solution, but couldn't find for n=3,4,5... This certainly validates the theorem but how do we prove it mathematically?

How did you "run a program" to check for all n= 3, 4, 5?

[CRGreathouse]

I wouldn't say it is "also known as". Yes, Wiles proved the Taniyama-Shimura conjecture. It has already been proven that Fermat's last theorem was true if and only if the Taniyama-Shimura conjecture was true.

I thought Taniyama-Shimura was stronger than Fermat's last theorem -- that Wiles proved special case of T-S first, enough to prove Fermat's last theorem, and later (with several others) proved the whole Taniyama-Shimura conjecture. Am I confusing this with something else?

[ron_jay]

It was bit of an accident that I stumbled upon Fermat's Last Theorem. I was originally coding a program to find the smallest number, which when squared, can be expressed as the sum of two different sets of individually squared numbers, which is:

65 = 1^2+8^2 = 4^2+7^2

in terms of three three numbers, you have:

325 = 1^2+18^2 = 10^2+15^2 = 6^2+17^2

in terms of four numbers, you have:

1105 = 23^2+24^2 = 4^2+33^2 = 9^2+32^2 = 12^2+31^2

This list goes on...but what I tried to do next is replace the power of n=2 with n=3 or greater integral values. What I encountered is that the program kept looping till infinity i.e. the program never terminated for the first set of two numbers cubed individually(a^3+b^n=c^n);this led me to doubt the program. I checked the program but found no errors. So I concluded that there was no solution for a power greater than 2. I incidentally also came across Fermat's last theorem.So my conclusion was indeed justified.

How did you "run a program" to check for all n= 3, 4, 5?

Well, here is a program in c++ for reference:

#include <iostream.h>
#include <conio.h>
#include <math.h>

int main()
{
long n=1,i,j,flag,s,k; //initializing
long p=3; //replace 'p' to check for any other power

while(1) //loop till infinity
{
flag=0;k=pow(n,p);

for(i=1;pow(i,p)<k;i++) //looping first number
{
for(j=i+1;pow(i,p)+pow(j,p)<=k;j++) //looping second number
{
s=pow(i,p)+pow(j,p);
if(s==k){flag++;}
if(flag==2){break;} //break out of loop if found
}
if(flag==2){break;}
}
if(flag==2){cout<<n;break;} //print number if found and the program terminates
n++;
}
return 0;
}

Replace 'p' for power of the expression, and you will find that for p=2, answer is 25 which rightly is the smallest pythagorean triplet; but for n>2 and n having an integral value, the program doesn't terminate i.e there is no solution.

[mgb_phys]

the program doesn't terminate i.e there is no solution.
Have you it for an infinite time yet?


To prove all odd numbers greater than 2 are prime numbers :
Mathematician: 3 is prime number, 5 is prime number, 7 is prime number, by induction, all odd numbers greater than 2 are prime numbers
Physicist: 3 is prime number, 5 is prime number, 7 is prime number, 9 is experiment error, 11 is prime number,......
Engineer : 3 is prime number, 5 is prime number, 7 is prime number, 9 is prime number, 11 is prime number,......
Computer Programmer: 3 is prime number, 5 is prime number, 7 is prime number, 7 is prime number, 7 is prime number,......
Statistician: Let us try some random numbers: 17 is prime number, 23 is prime number, 11 is prime number,......

[ron_jay]

ha...Alrighty. Agreed that the program will not run for an infinite time but if there was a solution to it, the program would have given it in a relatively short period of time.Wouldn't it? So you could conjecture that the equation is unlikely to have a solution.

[matt grime]

No. You cannot say that. The behaviour for a small, finite time, is no indicator of the over all behaviour.

[ron_jay]

No. You cannot say that. The behaviour for a small, finite time, is no indicator of the over all behaviour.

I don't mean to say that it is universally true that for a small behaviour, the overall behaviour is truly indicated. It was only in my case that the unavailability of the solution by the program for a small time led me to faintly suspect that there might be something peculiar about the solution. It was a thought pertaining only to the situation and was not a generalisation of sorts for all other mathematical concepts.

So what exactly is the Taniyama-Shimura conjecture?

[genneth]

See http://www.math.hmc.edu/funfacts/ffiles/20009.5.shtml

[matt grime]

If you're generalization to 'all integers' is based merely on some searhc not terminating, rather than examining the reason why it does not terminate quickly, then you've clearly done something rash.

[CRGreathouse]

ha...Alrighty. Agreed that the program will not run for an infinite time but if there was a solution to it, the program would have given it in a relatively short period of time.Wouldn't it? So you could conjecture that the equation is unlikely to have a solution.

Sure, you can conjecture that -- and you did, and you were right. There are lots of examples of problems with large smallest counterexamples, though, so matt took issue with your guess. Eh.

[ron_jay]

If you're generalization to 'all integers' is based merely on some searhc not terminating, rather than examining the reason why it does not terminate quickly, then you've clearly done something rash.

No. It's not rash at all.The search did not terminate for a reason, did it? And it is because according to the theorem there is no solution to an integral power above 2. So, the reason was examined and that is how I came across this ingenious theorem.

One way to go about solving it would be that a power of 4 for example can be expressed in terms of 2*2 or algebraically, 'p=mn'

(a^m)^n + (b^m)^n = (c^n)^n

Perhaps we could somehow use this identity(though I don't know how).

[mgb_phys]

Yes, as i said you don't have to check all infinite 'n' just the infinite number of prime 'n'.
Since any non-prime 'n' can be expressed as the sum of 2 prime, another tricky one to prove by the way!

[matt grime]

No. It's not rash at all.


yes it is, since you had no explanation of why it did not terminate,


The search did not terminate for a reason, did it?

what reason would that be? That you knew something about certain modular forms? Or because you didn't have a powerful enough computer, or didn't let your program run for long enough?


And it is because according to the theorem there is no solution to an integral power above 2. So, the reason was examined and that is how I came across this ingenious theorem.


that is complete BS. Example? OK, find an integer solution of

42323245673565835789467946805234545765831346256725 764x^6+ 13564683568357832346234567246895141234123545134876 54794679745x^5+26465324643257654876925634145232534 1654364575468798569869867357846986596234534x^2+x+1

or show none exists. Please, start your computer program now.....

[Dragonfall]

ha...Alrighty. Agreed that the program will not run for an infinite time but if there was a solution to it, the program would have given it in a relatively short period of time.Wouldn't it? So you could conjecture that the equation is unlikely to have a solution.

Let's say your program stops at 9,223,372,036,854,775,807 (if you're using a signed long variable in java), for n = 3. How do you know the solution isn't at 9,223,372,036,854,775,808?

On the other hand, if you can prove that your program will not terminate, then you can conclude that there are no solutions.

[mgb_phys]

On the other hand, if you can prove that your program will not terminate, then you can conclude that there are no solutions.
Unfortunately the halting problem is even less solved than fermat's last theorem!

[matt grime]

http://primes.utm.edu/glossary/page.php?sort=LawOfSmall

contains a couple of examples of the misleading behaviour of small numbers. I particularly like Skew's number. I remember another one about sums of powers as well, (perhaps the sum of 4 4th powers) that has only exceptionally large counter examples.

[robert Ihnot]

There was something called the Euler Conjeture: It takes n nth powers to make an nth power. This was largely accepted until it was shown--by virtue of a zero for the fifth left term, I have been told, that 27^5 + 84^5 +110^5 + 133^5 = 144^5. A larger case discovered in 1967 as well is: 85282^5 + 28969^5 +3183^5 + 55^5 = 85359^5. (It is interesting to note that 85359^5 = 4.53..x10^24, large enough to suite me.)

Also I am sure people know just as 3^2+4^2= 5^2, we have 3^3+4^3+5^3 = 6^3. Well it is sort of interesting that 4^5 + 5^5 +6^5 +7^5 +11^5 = 12^5.

[Dragonfall]

Unfortunately the halting problem is even less solved than fermat's last theorem!

He isn't asked to prove that his program will tell whether any other program will terminate. I'm sure there are proofs that uses the fact that an algorithm does not terminate. I can't think of anything non-trivial for the moment, but I remember seeing something like that in many proofs in graph theory.

[ron_jay]

yes it is, since you had no explanation of why it did not terminate,what reason would that be? That you knew something about certain modular forms? Or because you didn't have a powerful enough computer, or didn't let your program run for long enough? that is complete BS. Example? OK, find an integer solution of

42323245673565835789467946805234545765831346256725 764x^6+ 13564683568357832346234567246895141234123545134876 54794679745x^5+26465324643257654876925634145232534 1654364575468798569869867357846986596234534x^2+x+1

or show none exists. Please, start your computer program now.....

What you have said is perfectly correct, but the main purpose of my question has been defeated and it is not BS.I did not positively conclude that the anomaly of the program not terminating is caused due to Fermat's Last Theorem at first thought. It was only after learning about the theorem and then ruminating that perhaps, yes, the evidence that the equation does not have a solution as said by Fermat and then proved by Andrew Wiles may have an answer to to it not terminating, even if it is for a short period of time.Yes, I had beforehand knowledge that there may not be a solution to the equation and hence for any great value(9,223,372,036,854,775,807), the program will not return an answer. Unfortunately, lot of misunderstanding has crept in.The program does not prove or validate Fermat's Last theorem(that's were I think we are going wrong),but is in fact a consequence of it.

[matt grime]

From post 1 by ron_jay:

I recently ran a program to find this solution, but couldn't find for n=3,4,5... This certainly validates the theorem but how do we prove it mathematically?

from post 28 by ron_jay:

the program does not prove or validate Fermat's Last theorem(that's were I think we are going wrong),but is in fact a consequence of it.

That's where we are going wrong???

[ron_jay]

Let's say your program stops at 9,223,372,036,854,775,807 (if you're using a signed long variable in java), for n = 3. How do you know the solution isn't at 9,223,372,036,854,775,808?

On the other hand, if you can prove that your program will not terminate, then you can conclude that there are no solutions.

Correct, that if you can prove that the program does not terminate, the theorem will be proved just like the Taniyama-Shimura Conjecture was a precedent to the proof of Fermat's Last theorem.Right, we don't know whether the last possible long digit available to the program is the solution or not and it could be, but all I am doing is laying down a conjecture that if the program had not terminated and we knew that it hadn't in some way, we would arrive at a conclusion and you have rightly voiced my own opinion in every way.

Let's say I name it the "NON-Terminating Program Conjecture".:smile:

[ron_jay]

That's where we are going wrong???

Yes, on that post I hadn't elucidated the exact details(how misleading words can be!) and hence we went wrong in not understanding what I meant(my mistake for not expanding the first post) and went on to talk about the program instead of how we could actually prove the theorem mathematically.

[Dragonfall]

I think by "validates" he meant "correlates", in the first post.

[learningphysics]

ron_jay was just saying that he conjectured... he made a guess... Then he looked up Fermat's last theorem and his guess turned out right. It was just supposed to be about how he came across FLT... not that he had justified it or validated it.

Whether or not the guess is unjustified, the fact is that the guess turned out to be true... Do all guesses need to have some sort of proper justification? It's just a guess after all...

[Gib Z]

find an integer solution of

42323245673565835789467946805234545765831346256725 764x^6+ 13564683568357832346234567246895141234123545134876 54794679745x^5+26465324643257654876925634145232534 1654364575468798569869867357846986596234534x^2+x+1

or show none exists.

It seems that is not an equation :(

Sometimes too much mathematical rigor destroys mathematical intuition, which in my opinion is more important. As robert_ihnot pointed out, even Euler has been guilty of the mistake: Not seeing any counter examples, so believes in the theorem for all examples. It is true, these days with the ever more complex maths, large counter examples are becoming more common, and matt grime, being a modern day mathematician, has good reason to be careful of them. He is merely trying to teach others to be the same.

[matt grime]

Ok, root, then.

[Gib Z]

This is getting away from the original point, but x=0 is a solution >.<

[matticus]

when x = 0 the equation is 1.
But as far as the purpose of this thread, to explain wiles proof, i don't think many of us on here would get much out of it. when wiles made the error in his proof initially he said "Even explaining it to a mathematician would require the mathematician to spend two or three months studying that part of the manuscript in great detail."

[ron_jay]

when x = 0 the equation is 1.
But as far as the purpose of this thread, to explain wiles proof, i don't think many of us on here would get much out of it. when wiles made the error in his proof initially he said "Even explaining it to a mathematician would require the mathematician to spend two or three months studying that part of the manuscript in great detail."

Sure, you are correct that this kind of proof is not easy to understand and I totally agree with Wiles comment, but that exactly is the purpose of this thread - to prove that theorem...even if it takes years, no problem...I believe that when many heads come together for a common purpose they can achieve things bigger than miracles...

[matticus]

the theorem has already been proved. are you looking for an alternate proof?

[ron_jay]

Yes, without referring to what Wiles did and the possible lines along which we could proceed will be the first step.

[HallsofIvy]

You understand, do you not, that people looked for a proof for almost 500 years before Wiles came up with his! If you are expecting a different proof, "even if it takes years" may not be enough! And certainly, any different proof will be no easier to understand than Wiles'.

[CRGreathouse]

And certainly, any different proof will be no easier to understand than Wiles'.

I don't think that's certain at all. I would be disappointed if the mathematical community doesn't find a more 'elegant' solution in the next 500 years -- though it will probably use concepts much more advanced than we have today (being thus less 'elementary'), it would be 'simpler' in terms of chunking and perhaps even elucidate more.

(I would also hope for such an improved proof of the four-color theorem, but that may never happen -- somehow I see Wiles' theorem as more fundamental and more worthy of continued effort.)

[Dragonfall]

Wouldn't it be great if you could prove that a proof exists?

[matticus]

what's the difference between proving that a proof exists and proving the theorem? if you prove that a proof exists then assuming the theorem was false would lead to a contradiction. so saying a proof exists is essentially proving it isn't it?

[Moridin]

what's the difference between proving that a proof exists and proving the theorem? if you prove that a proof exists then assuming the theorem was false would lead to a contradiction. so saying a proof exists is essentially proving it isn't it?

To prove a statement in mathematics means that you combine established axioms via established means of inference and reach that statement. It is like saying 'provided that the axioms apply, the theorem apply as well'.

[ron_jay]

yes,its like a stack of books.If the the lower books hold, then the upper books will also hold .Already established axioms are like the lower books and the newer proofs depend on them.

[matticus]

i thought i understood proofs, i've done many of them (though i am still doing my undergraduate work).

by saying a proof exists, we are saying that the conjecture is valid, are we not? that is what i take it to mean.

if there is a proof to a conjecture, then a conjecture is valid.
there is a proof to the conjecture
therefore the conjecture is valid.

so by using that logic we have proved the validity of the conjecture, and yet that is not enough?

[matt grime]

Knowing that something exists is different from a construction that demonstrates it. Many things in mathematics are non-constructive. OK, I can't think of any examples where one knows that a proof exists, but one cannot write it down, but I'm no logician.

[matticus]

so are you agreeing with me that knowledge of a proof would constitute a proof despite there not being a construction? or must the construction exist? if someone proved there was a proof to the Riemann hypothesis, would mathematicians now be justified to say "as we now know R.H. is true..."? this whole thing is really kind of stupid, but i think it would be interesting to see any example where a proof is known to exist and yet has not been found, simply because the idea is so counterintuitive.

[Dragonfall]

You normally "prove the existence of a proof" by writing the proof down. I've never seen a proof a la "suppose a proof does not exist", but I imagine something like that could exist in higher order logic or something.

[matt grime]

I am neither agreeing nor disagreeing with you, matticus. The question is so ambiguous as to be unanswerable.

[HallsofIvy]

Here is a proof that comes awfully close to that:

Prove the there exist irrational a, b such that ab[/sub] is rational.

Look at \sqrt{2}^\sqrt{2}[/itex]. It is not known (last time I checked) if that number is rational or irrational. However:

If it is rational, then we are done.
If it is irrational, then [tex]\(\sqrt{2}^\sqrt{2}\)^\sqrt{2}\)= \sqrt{2}^2= 2 is rational.

We have proved that there exist irrationals a, b such that a[sup]b is rational but are unable to say what a and b are!

[matticus]

yeah that proof was in my discrete math book, it is cool. the example i was going to use was that given any number we know there is a bigger prime, despite the fact that at some point we won't be able to construct it. neither of these are the same thing, but i think one of the things that attracts me to math is that you can know things exist without having to see them exist. where in science statistical evidence is very important, in math it's almost irrelevant.