# Fermat's Last theorem

#### 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.

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#### 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

Erroneous

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

42323245673565835789467946805234545765831346256725764x^6+ 1356468356835783234623456724689514123412354513487654794679745x^5+264653246432576548769256341452325341654364575468798569869867357846986596234534x^2+x+1

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

Homework Helper
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". #### 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

Homework Helper
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

Homework Helper
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.

Homework Helper
Ok, root, then.

#### Gib Z

Homework Helper
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

Homework Helper
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

Homework Helper
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

Homework Helper
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.