Fermat’s Last Theorem

Written by on 27 April 2011

Topics: Math

Fermat’s last theorem is one of the best known mathematical puzzles ever posed. It is very easy to understand yet it eluded a proof for 350 years. Fermat stated in the margin of Arithmetica that he had the most marvellous proof of the conjecture, but it was too long to fit in the margin. It has always been known as Fermat’s last theorem even though it has only been a conjecture for 350 years. Pierre de Fermat stated that

it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.

In other words

does not have solutions for n > 2.

For n = 2 there exist infinitely many solutions and we have been dealing with them in problem 9 of Project Euler.

The problem was finally solved in 1995 by Andrew Wiles after he dedicated 8 years struggling to prove the theorem. In order to prove the theorem he had to prove several other conjectures and not least use results and methods in many branches of mathematics developed within the last 100 years. So there is no way that Fermat could have proven it the same way as Andrew Wiles did. Wikipedia has a section on Fermat’s Last Theorem where they briefly go through the history and the content of the proof.

My story on the theorem

I was first introduced to Fermat’s last theorem when I went in high school a mere 4 years after the theorem of proven. Our math teacher (whom I owe a lot of thanks for sparking my curiosity) wanted us to watch the movie on the subject made by Simon Singh and John Lynch. Many of my fellow students giggled at Andrew Wiles and thought he was a complete nut job, but I saw something different. I saw a man with a burning passion for solving this problem, and by the end of the movie I was so touched that I was almost crying. To me it was a real story and a treasure hunt for the truth.

This movie sparked something in me, and inspired me in many ways. I don’t claim to be good at mathematics and I am not rigorous enough to prove many things. But my passion and curiosity for math was ignited and will burn forever after this movie. It has been aired on television in many countries and until Google video was closed it was available through that service. Today it might be available through other means on the internet, but I haven’t found a source for it. If you a legal source for the movie I would be very interested in hearing from you.

Simon Singh is also the author of a book on the subject called Fermat’s Last Theorem. Simon Singh is a great story teller and manages to take the reader through the story in a way that most people can follow. The book takes you all the way from Fermat’s life and achievement and through the long history of Fermat’s Theorem which many people have spend many hours trying to prove without success until Wiles finally did his brilliant work and finally proved it. So if you love a good story I will highly recommend you to read the book and watch the movie.

On a side note Simon Singh has written other great books on different subjects such as The Code Book.

But did he prove it?

The big question is; did he prove it? Not Wiles of course but Fermat – did he prove it? Most sources believe not. Wolfram has the very good argument that he later on looked for proofs of n=4 and n=5, which would have been meaningless if he had already proven it.

Due to my personal pride I hope and doubt he in fact did not prove it, because that means he would have had an insight that has eluded the rest of humanity for 350 years even though the mathematics has evolved incredibly since then.

19 Comments For This Post I'd Love to Hear Yours!

  1. w.johnson says:

    I think Fermat did come up with the proof. It was probably something like this: Fermat’s equation is simply the Pythagorean identity times some factor c to the n. Sine squared theta and cosine squared theta are irrational via Niven’s theorem. So a and b to the n are irrational so they can’t be integers.

  2. Why Wiles’ proof need 100 pages. I cannot even under the first page.
    Here is a proof I could understand:
    A^n+B^n=C^n
    If A^n and B^n are general symmetrical twin-primes, then globally:
    1/2*(A^n+B^n)=C^n
    But both sides must be integeral. Therefore
    C = ( 1/2*(A^n+B^n))^(1/n)
    But the right-side is impossible since the nth root of 2 is irrational. Therefore FLT is proved globally. Q.E.D

  3. Corrections:

    Why Wiles’ proof need 100 pages. I cannot even under the first page.
    Here is a proof I could understand:
    A^n+B^n=C^n
    If A^n and B^n are general symmetrical twin-compositess, then globally:
    1/2*(A^n+B^n)=C^n
    But both sides must be integeral. Therefore
    C = ( 1/2*(A^n+B^n))^(1/n)
    But the right-side is impossible since the nth root of 2 is irrational. Therefore FLT is proved globally. Q.E.D

  4. Kristian says:

    Without knowing what you mean with general symmetrical twin-composites, I really doubt that the proof holds. In fact as far as I can see, your assumption that you can pull out 1/2, shows an assumption you cannot make. And therefore your proof does not hold.

    And if you really are correct, I think you should publish that rather in a peer-reviewed journal rather than here, because that would be a really feat to find such a simple proof for something that have puzzled mathematicians for 300 years.

  5. Here is my short proof of Fermat’s Lsst Theorem, please critize if
    found wrong.

    Aditi Journal of Computational Mathematics
    Volume 1 ( 2013 ) , Issue 2
    Previous | Next | Back

    A Short Proof of Fermat’s Last Theorem
    Huen Yeong Kong
    Pages 1-4
    View Details Abstract References

    The paper starts with a linear equation c = (a+b) where c, a and b are positive integers. Global integral equality is assured in this equation. Raising both sides of this equation to the nth power we get cn = (a+b)n which still retains global integral equality. If the right-side is expanded as Binomial Theorem, we get cn = an + unexpanded intermediate bionomial terms + bn. If the intermediate binomial terms could be reduced to zero, we get Fermat’s Last Theorem. But this is an impossibility since the right-side is uniformly additive with a and b as positive integers and n>2. This proves Fermat’s Last Theorem using only 17th centuary mathematics.

  6. Kristian says:

    You are right that if we get that all the intermediate terms can be removed we have Fermat’s last theorem. However, you have not shown that there isn’t a way to group all these terms into the form a^n + b^n. And therefore your proof does not hold in general.

  7. Huen Yeong Kong says:

    I think you have a point. I will think about it. Meanwhile if
    anyone could improve on my paper, that would be welcome. Thanks
    for the advice.

    Huen Yeong Kong, Singapore

  8. Here is my attempt at an alternative proof of FLT:
    For integral equality of FLT: z^n=x^n+y^n, let x=y then
    z^n=2*x^n
    Then z must be equal to 2^(1/n)*x for integral equality.
    However 2^(1/n) is irrational, therefore 2^(1/n)*x is also irrational. Therefore there is no integral equality for FLT
    based on FLT assertions.

    Huen Yeopng Kong

  9. Kristian says:

    Hi Huen

    Yep that would work for the special case where x = y, now you just need to show it for the infinity of cases where x != y.

    I am sorry to sound rude, but I think you should stop these attempts. You wont find a proof for FLT like this. You should rather use your energy and skills on solving some other problems. I can’t guide you in the right direction here, you would need a mathematician for that.

  10. Huen Yeong Kong says:

    Yes, you are right. Should not spend too much energy on this topic now. But I did learn a lesson on Fermat’s strategy in setting up his conjecture. Start with a globally well-behaved equation, expand it,
    and drop part of it. Then present the deformed formula to the world. It took 370+ year before Wiles solved his problem. Have a good day.
    Huen Yeong Kong

  11. Kristian says:

    Yes, that is a very good point. Start from something you know for certain and then work towards what you want to prove. That is always a good approach.

    Not the only approach, but a good one.

  12. Elementary number theory is simple. School leavers and amateur number theorist like me could comprehend statements of FLT and Goldbach’s Conjecture almost instantly. With modern number theory, things are not so simple. This group has grown inward cutting themselves off from school leavers and amateur number theorists. I am 82 years old. If I am younger I would like to be an activist to revive interest in elementary number theory. Even Wolfram gives only a scant one line remark on elementary number theory. There are still plenty of interesting ideas coming out from elementary number theory.
    Huen Yeong Kong

  13. R.A.D.Piyadasa says:

    I believe Fermat had a proof when he he made his famous statement that he had a marvelous proof. But he might found a mistake in it later.I believe his proof is based on a parametric solution of the equation of the theorem x^n+y^n=z^n.Still I could not derive what he had in his mind but the one ‘A simple and short analytical proof of Fermat’s last theorem’ which one can read on the internet(Published in CMNSEM)is close to it,to my mind.Within a short period of time I hope to derive a much shorter one which I believe the one that Fermat had in his mind.

  14. In other words

    \displaystyle a^n b^n = c^n

    does not have solutions for n > 2.

    For n = 2 there exist finitely many solutions

    I think you mean *infinitely* here.

  15. Kristian says:

    Yes I do, thanks for noting.

  16. Kumaresan K says:

    Thanks,I wish to share the following reg FLT. Consider any two positive
    integers, I have taken 9 & 10, only a random choice.
    9^1+10^1=19
    9^2+10^2=181 -> (181)^1/2 a irrational number between 13 & 14,
    9^3+10^3=1729 -> (1729)^1/3 a irrational number between 12 & 13,
    9^4+10^4=16561 -> (16561)^1/4 a irrational number between 11 & 12,
    9^5+10^5=159049 ->(159049)^1/5 a irrational number between 10 & 11,
    Now what about nth root of 9^n+10^n for n>5?. The sum 9^n+10^n converges
    to 10^n and nth root lies between 10 & 11, all are irrational Nos.
    FLT for 9 & 10 verified.

  17. Pham Duc Sinh says:

    1. There is another explanation of a simple proof of Fermat’s last theorem as follows:

    X^p + Y^p ?= Z^p (X,Y,Z are integers, p: any prime >2) (1)

    2. Let‘s divide (1) by (Z-X)^p, we shall get:

    (X/(Z-X))^p +(Y/(Z-X))^p ?= (Z/(Z-X))^p (2)

    3. That means we shall have:

    X’^p + Y’^p ?= Z’^p and Z’ = X’+1 , with X’ =(X/(Z-X)), Y’ =(Y/(Z-X)), Z’ =(Z/(Z-X)) (3)

    4. From (3), we shall have these equivalent forms (4) and (5):

    Y’^p ?= pX’^(p-1) + …+pX’ +1 (4)
    Y’^p ?= p(-Z’)^(p-1) + …+p(-Z’) +1 (5)

    5. Similarly, let’s divide (1) by (Z-Y)^p, we shall get:

    (X/(Z-Y))^p +(Y/(Z-Y))^p ?= (Z/(Z-Y))^p (6)

    That means we shall have these equivalent forms (7), (8) and (9):

    X”^p + Y”^p ?= Z”^p and Z” = Y”+1 , with X” =(X/(Z-Y)), Y” =(Y/(Z-Y)), Z” =(Z/(Z-Y)) (7)

    From (7), we shall have:

    X”^p ?= pY”^(p-1) + …+pY” +1 (8)
    X”^p ?= p(-Z”)^(p-1) + …+p(-Z”) +1 (9)

    Since p is a prime that is greater than 2, p is an odd number. Then, in (4), for any X’ we should have only one Y’ (that corresponds with X’) as a solution of (1), (3), (4), (5), if X’ could generate any solution of Fermat’s last theorem in (4).

    By the equivalence between X’^p + Y’^p ?= Z’^p (3) and X”^p + Y”^p ?= Z”^p (7), we can deduce a result, that for any X” in (8), we should have only one Y” (that corresponds with X’’ ) as a solution of (1),(7),(8),(9), if X” could generate any solution of Fermat’s last theorem.

    X” cannot generate any solution of Fermat’s last theorem, because we have illogical mathematical deductions, for examples, as follows:

    i)In (8), (9), if an X”1 could generate any solution of Fermat’s last theorem, there had to be at least two values Y”1 and Y”2 or at most (p-1) values Y”1, Y”2,…, Y”(p-1),
    that were solutions generated by X”, of Fermat’s last theorem. (Please note the even number (p-1) of pY”^(p-1) in (8)). But we already have a condition stated above, that for any X” we should have only one Y” (that corresponds with X”) as a solution of (1),(7),(8),(9), if X” could generate any solution of Fermat’s last theorem.
    Fermat’s last theorem is simply proved!

    ii)With X”^p + Y”^p ?= Z”^p, if an X”1 could generate any solution of Fermat’s last theorem, there had to be correspondingly one Y” and one Z” that were solutions generated by X”, of Fermat’s last theorem. But let’s look at (8) and (9), we must have Y” = -Z”. This is impossible by further logical reasoning such as, for example:

    We should have : X”^p + Y”^p ?= Z”^p , then X”^p ?= 2Z”^p or (X”/Z”)^p ?= 2. The equal sign, in (X”/Z”)^p ?= 2, is impossible.
    Fermat’s last theorem is simply again proved, with the connection to the concept of (X”/Z”)^p ?= 2. Is it interesting?

  18. Kristian says:

    I must admit that I don’t follow you all the way through, but just the fact that you assume that p is a prime means that you have not proven Fermat’s last theorem. You might have proven something interesting, but FLT it is not.

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