Fermat's Last Theorem: Proving the Impossibility of Integral Solutions for n>2

[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

42323245673565835789467946805234545765831346256725764x^6+ 1356468356835783234623456724689514123412354513487654794679745x^5+264653246432576548769256341452325341654364575468798569869867357846986596234534x^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 a<sup>b[/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:<br /> <br /> If it is rational, then we are done.<br /> If it is irrational, then \(\sqrt{2}^\sqrt{2}\)^\sqrt{2}\)= \sqrt{2}^2= 2 is rational.<br /> <br /> We have proved that there exist irrationals a, b such that a^b is rational but are unable to say what a and b are!</sup>

 

[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.