1. ## Fermat's Last Theorem

There is an old saying which goes, you can never prove a negative. In other words you can only prove what will happen not what won't happen.

In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two.

So how do you go about proving this? Andrew Wiles did it in 1994 but how did he prove a negative?

2. Originally Posted by Just Curious
... you can never prove a negative.
(emphasis mine)

Isn't it a negative statement?

Jokes aside, of course you can prove a negative. For example, there is no real solution to the equation

$x^2+1=0$

More generally, a "negative" is only negative because it is the negation of another statement, but the original statement is "negative" relative to its "negative". I think you wanted to say nonexistence.

3. Temur. I was thinking more philosophically at first.

In general, statements of the form "X exists" are (if true), easy to prove. One simply shows an example of X as a proof. For example, I can prove a claim that "White swans exist" by exhibiting a normal (European) swan. I can similarly prove that "black swans exist" by exhibiting an Australian one.

On the other hand, it is in theory impossible to prove beyond question that "No green swans exist" (or more generally any statement of the form "No X exists"). I can show cages and cages full of five thousand swans, all of which are white and black, but what about the five thousand and first? In order to prove that no green swans exist, I would need to produce and examine every swan in existence, including swans on unexplored Pacific islands and even unknown planets in distant galaxies

This URL goes into it more deeply.
http://skepticwiki.org/index.php/%22..._a_Negative%22

The point here releated to Fermat is that nobody can test every single combination of numbers to prove that none exist which meet the criteria he conjectures.

4. Swans are things in real worlds so it is (almost) impossible to prove there is no green swan. But solutions to equations are logical constructs with strict mathematical definition, so it is possible (at ;east in principle). This can be done by first assuming there exists a solution and arriving at a contradiction, or by testing every single combination of numbers into the equation (this is possible!). To see how the latter works, consider the equation

$x^2+1=0$

For every real $x$, $x^2\geq0$, so we always have $x^2+1\geq1>0$.

Have look at

http://www.tricki.org/article/Imposs...nce_front_page

5. Originally Posted by Just Curious
There is an old saying which goes, you can never prove a negative. In other words you can only prove what will happen not what won't happen.

So how do you go about proving this? Andrew Wiles did it in 1994 but how did he prove a negative?
Plenty of things in mathematics are proofs of somethings non-existence. Proof by contradiction is one way, where you assume something does exist and then prove it leads to obviously nonsense results, like 1=2. Other times its just straightforward. For instance, I can prove there is no even prime number larger than 2. Suppose there is, call it p. p is even, therefore p = 2q for some natural number q>1. But 2q has a non-trivial prime factorisation, as its divisible by 2. Thus p cannot be prime. Thus there exists no even prime p>2.

In physics you can't prove a negative because it'd basically require you to search the entire universe, as you're having to discover if something is or isn't there. In mathematics you know all the relevant rules of the game and proving something doesn't exist is like proving the rules don't allow it.

/edit

By the way, the proof that Wiles devised is staggeringly complicated. Even with a PhD in mathematics under your belt you'd not know enough to follow his entire proof. I know he used things like Galois theory and last week I spoke to someone who did a PhD in Hopf-Galois theory, particularly field extensions and modular forms (the sort of things in the proof) and he didn't follow the proof (it came up in discussion while he explained his thesis to a couple of people). The common 'internet hack' attempts at FLT focus in basic mathematics but the actual proof is far far beyond that, beyond almost anyone. I know of Galois theory and I know (or used to know) what a field extension is but beyond that I am in over my head (yes, despite what the hacks here think, I do admit my limits). There's definitely people with much deeper knowledge on those things here but (and I mean no offence to such people, like Temur) I would be very surprised if even they could grasp a significant chunk of Wiles' work.

6. No, I don't grasp Wiles' work in any detail. There are expositions on the high level structure of the proof but then you have to take lots of things on faith. For me Perelman's work is much more accessible than Wiles'.

7. Thanks guys, I understand now the difference between real world problems and mathamatical proofs.

On the subject of Wiles, his dedication to finiding the proof of FLT is as staggering as the proof itself. After 7 years of work on the first proof, to then be told there was an error, would have knocked the stuffing out of most people, but he went on and a year later solved it. This is inspiration for anybody working on any sort of problem, not just in maths.

8. Originally Posted by Just Curious
Thanks guys, I understand now the difference between real world problems and mathamatical proofs.

On the subject of Wiles, his dedication to finiding the proof of FLT is as staggering as the proof itself. After 7 years of work on the first proof, to then be told there was an error, would have knocked the stuffing out of most people, but he went on and a year later solved it. This is inspiration for anybody working on any sort of problem, not just in maths.
I think you might view the documentary film about how Andrew Wiles could prove the Fermat's last theorem. (infocobuild.com/books-and-films/science/fermat-last-theorem.html)

9. You prove a negative by Reductio ad absurdum.

We will prove Archimedes assertion that there does not exist a greatest positive integer.

Let k be that greatest natural number.

Take k + 1.

k + 1 > k.

Hence, these does not exist a greatest positive integer.

We proved a negative.

10. Originally Posted by Jack_
We will prove Archimedes assertion that there does not exist a greatest positive integer.

Let k be that greatest natural number.

Take k + 1.
How can you take k+1 when k is supposed to be the greatest natural number? You're assuming what you're trying to prove, which is a no-no.

Try again.

11. Originally Posted by James R
How can you take k+1 when k is supposed to be the greatest natural number? You're assuming what you're trying to prove, which is a no-no.

Try again.
LOLOLOOL

This the historical proof.

I take K + 1 because that is a property of peano arithmetic.

http://en.wikipedia.org/wiki/Peano_axioms

For every natural number n, S(n) is a natural number.

What I presented is simple logic.

I do not know how to simplify it further.

12. Originally Posted by Jack_
I take K + 1 because that is a property of peano arithmetic.

http://en.wikipedia.org/wiki/Peano_axioms

For every natural number n, S(n) is a natural number.

What I presented is simple logic.

I do not know how to simplify it further.

So what you have proved is: assuming that the Peano axioms hold, there is no greatest integer.

You have no general proof.

13. Originally Posted by James R

So what you have proved is: assuming that the Peano axioms hold, there is no greatest integer.

You have no general proof.
I cannot even believe you do not understand this method of proof.

Let's get down to child's play.

Does there exist a greatest integer yes or no.

14. Originally Posted by Jack_
I cannot even believe you do not understand this method of proof.
Do you not understand that you made some unstated assumptions?

Let's get down to child's play.
No thanks. You can go play with the kids.

Does there exist a greatest integer yes or no.
Under what assumptions/axioms?

15. Originally Posted by James R
Do you not understand that you made some unstated assumptions?

No thanks. You can go play with the kids.

Under what assumptions/axioms?
Under what assumptions/axioms?
Peano arithmetic.

That should have been obvious given Archimedes argument.

My logic will not be refuted.

16. Originally Posted by Jack_
Under what assumptions/axioms?
Peano arithmetic.
Fine. You should have said that in the first place.

Are we done?

17. Originally Posted by James R
Fine. You should have said that in the first place.

Are we done?

Yea, as long as you agree I am correct.

18. What you have proved is: assuming that the Peano axioms hold, there is no greatest integer.

That is correct. Done?

19. Originally Posted by James R
What you have proved is: assuming that the Peano axioms hold, there is no greatest integer.

That is correct. Done?
If you do not agree with Peano axioms then you cannot possibly agree with any conclusions from physics.

If so, state how.

20. It's ok, Jack_. I believe the Peano axioms are a reasonable basis for arithmetic.

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