# Simple solution of FLT?

1. Jul 5, 2005

### vantheman

Am still trying to find a simple solution of FLT. I think I remember that someone proved that if a solution exist to FLT that either x, y or z must have a factor of n. Is this true and, if so, where can I find the proof of this?

2. Jul 5, 2005

3. Jul 5, 2005

### Zurtex

I'm assuming you mean Fermats Last Theorem and not Fermats Little Theorem.

The theorem states that there exits no solution for:

$$\left. x^n + y^n = z^n \quad \forall \, x, y, z, n \in \mathbb{N} \right\backslash \{ 1 \}$$

And it has been proven true.

4. Jul 5, 2005

### vantheman

Yes, I was referring to Fermat's Last Theorem. I know Andrew Wile has already solved it, but there is that remote possibility that Fermat's "truely marvelous proof" does exist.
It may seem like a waste of time to a serious mathematician, but amuse me if you will.
I don't understand your reply because I do not have the mathmetical credentials. Let me re-phrase the question. Prior to Wile's proof, did anyone prove that, in order for a solution to exist, either x, y or z must have a factor of n?

I don't feel my efforts have been wasted. I have developed an independent proof for n=3 using simple algebra and congruences. It has been reviewed by the Math Dept of a local university and sent to the College Mathmetical Journal for possible publication.
If the above proof exists, it will go a long way to extending the proof for n=3 to be a proof for all prime values of n. HELP.

5. Jul 5, 2005

### HallsofIvy

Staff Emeritus
I presume you meant $$n \in \mathbb{N}\right\backslash\{1,2\}$$

6. Jul 5, 2005

### HallsofIvy

Staff Emeritus
Do you mean this n to be the same as the power in xn+ yn= zn?

Are you sure of that? The cases n= 3, 4, 5 were quickly proved but the method used did not extend to n larger than 5.

7. Jul 5, 2005

### vantheman

It's Greek to me.

Yes, my Greek is a little rusty.

8. Jul 5, 2005

### Icebreaker

I'm sure Fermat actually wrote, "Wouldn't it be funny if I have a truly marvelous proof..." and the first part was torn away somehow. :rofl:

9. Jul 5, 2005

### vantheman

Let me state it again:
Prior to Wile's proof, did anyone prove that in order for a solution to exist to the equation
x^n + y^n = z^n in non-zero integers
that either x, y or z must have a factor of n?

Also, since my proof for n=3 has the possibility, through binomial expansion techniques, to be a tool for proving the equation for all prime values of n, an existing proof of the question I posed will help me in extending my proof for n=3 to n = all prime numbers.

10. Jul 5, 2005

### Zurtex

You are of course right, doh!

I've never heard of that before. In fact that sounds weird, because the trouble with the methods of proving it for specific cases only struggled on certain prime numbers. I think composite numbers were really easy to deal with, before Wile's proof I think it was something like all n < 10'000'000 were proven.

11. Jul 5, 2005

### matt grime

n=3 and n=4 are easy to do by elementary methods, 4 is easier than 3 if i recall correctly.

there are many things out there in number theory (not to mention other parts of mathematics) that are interesting and have open questions to work towards, and indeed studying the number theory of various things to do with FLT can be rewarding but don't bother trying to get an elementary proof of FLT as there is almost certainly no such. There have been many plausible attempts that failed. some for example because they assumed unique factorization in rings that turned out not to be UFD's. in fact there are probably sufficiently many "proofs" out there that Fermat's original "proof" has been discovered, and is incorrect. Fermat was a genius but like lots of us he got many things wrong you'd be surprised how many old well known results attributed to the greats were never properly proven by them. it took some 40 years for euler to anwer some simple questions of fermat's never mind the complicated ones (relatively speaking that is).

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html

contains the information you want (amazing what google can do: google for fermat's last theorem known conditions)

Last edited: Jul 5, 2005
12. Jul 5, 2005

### vantheman

Thanks all. Have to run now. Will check back later.

13. Jul 5, 2005

### matt grime

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html

i case you didn't read my edited post has all the answers you need, i imagine. but why not just study number theory in general.

14. Jul 5, 2005

### shmoe

Germain showed this is the case if n and 2n+1 are prime. Actually she proved something a little more general have a look for Sophie Germain's theorem or Sophie Germain primes. I've never seen a full generalization that holds for all primes n.

15. Jul 6, 2005

### keebs

A simple solution for FLT would arise if you could prove the abc conjecture...

16. Jul 7, 2005

### shmoe

...in a simple way

17. Jul 8, 2005

### robert Ihnot

VAntheman: I think I remember that someone proved that if a solution exist to FLT that either x, y or z must have a factor of n. Is this true and, if so, where can I find the proof of this?

Here is where it's at: We only need consider prime cases, 4 was handled by Fermat, and we look at:

X^P+Y^P=(X+Y)(X^(P-1) -X^(P-2)Y+-....Y^(P-1)) = Z^P

If X+Y is divisible by a factor of Z, call it M; then X==-Y Mod M, so that:

X^(P-1) -X^(P-2)(-X) +X^(P-3)(-X)^2 -+-==PX^(P-1), Mod M. But X, Y and Z are presumed to have no common factor, so the only factor that could be in common with the factorization is P. Other than that we have both sides of the factorization are perfect Pth powers. That is how we have classically divided the two cases, Case 1: P divides one of the terms, or Case 2: P does not.

I don't think the above proves it is necessary for P to divide one of X,Y, or Z. HOWEVER, case 1 historically is much easier to eliminate for a given P, which would say that for a solution, P would have to divide one of the terms. BUT THERE ARE NO SOLUTIONS, so yes! P would have to divide one of the terms for there to be a solution.

Last edited: Jul 9, 2005
18. Jul 9, 2005

### robert Ihnot

HallsofIvy: The cases n= 3, 4, 5 were quickly proved but the method used did not extend to n larger than 5.

Hold on here! N =4 was presented by Fermat himself, one of the few things he did prove, and by the method of infinite descent. Euler is credited with the proof for n=3 in 1700, but Dirichlet did not prove the case for n=5 until 1828. Next case n=7 was 7 years later by Gabriel Lamé and Henri Lebesque.
http://www.bath.ac.uk/~ma1jam/Fermat/ [Broken]

Last edited by a moderator: Apr 21, 2017 at 5:33 PM
19. Jul 9, 2005

### vantheman

Thanks, but this is a special case where x + y is factorable. If it is prime then it doesn't work. I was hoping for a general solution. I guess it doesn't exist.

20. Jul 9, 2005

### mathwonk

isn't it more likely that fermat was mistaken, than that 350 years of mathematicians have failed to find his elementary proof?