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Posers and Puzzles

Posers and Puzzles

  1. 02 Sep '04 07:32
    Is anyone here familiar enough with Fermats last theorem
    to help me understand why its so difficult to prove?
    What did it take? Like 600 years?
    I've read some texts on the subject but they all go
    very high above my head. Can someone here maybe put it
    to me in less technical language?
    (I figure this a better place than some pure math forum since
    I probably wouldnt understand their answers)
  2. Standard member TheMaster37
    Kupikupopo!
    02 Sep '04 09:13
    What i know of it (and that isn't much) is that everal new branches of math has to be discovered before it could be proven. Furthermore it required to cooperation of alot of different theorems.Some of those were big. The complexity of the proof was so great that a single person couldn't get it to work, but several persons found bits and pieces, until someone realised that the last step in completing the proof was found.
  3. 02 Sep '04 10:41
    Originally posted by TheMaster37
    What i know of it (and that isn't much) is that everal new branches of math has to be discovered before it could be proven. Furthermore it required to cooperation of alot of different theorems.Some of those were big. The complexity of the proof was so great that a single person couldn't get it to work, but several persons found bits and pieces, until someone realised that the last step in completing the proof was found.
    Yes, from what I gather the proof is infinitly complex.
    I was just wondering if there is a easier way to understand
    why it was so difficult. If maybe there is some easily understood
    reason why less complicated attempts fail?
    There may not be. I just thought I'd ask :-)
  4. 02 Sep '04 23:17
    Fermat's Last Theorem didn't take 600 years to prove, only about 335. The problem is that the theorem covers an infinite number of cases, so if it can't be proved by induction, then you probably have to find a very roundabout wayof proving it. An even older example is the theory that their are infinitely many numbers of prime pairs. It is generally regarded as probably true, but the problem of proving it has remained unsolved for over 2000 years.
  5. Standard member TheMaster37
    Kupikupopo!
    03 Sep '04 14:41
    Wasn't that problem recently solved?
  6. 04 Sep '04 18:05
    No, it's still unsolved, but there has been a recent discovery of a method of finding prime numbers which are 'unusually close', though not necessarily pairs.

    see http://news.bbc.co.uk/2/hi/science/nature/2911945.stm
  7. 08 Sep '04 13:17
    Going back to Fermats theorem, the problem itself it really easy to understand, its the solution which is the problem

    For reference and because its not been included abouve in this thread the problem is this:

    Prove that xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2

    Because of the >2 element there are infinity options to factor which under current thinking at leat is impossible and so proving definitively that the statement is true requires a workaround of mathematics.

    The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA. Wiles gave a series of three lectures at the Isaac Newton Institute in Cambridge, England the first on Monday 21 June, the second on Tuesday 22 June. In the final lecture on Wednesday 23 June 1993 at around 10.30 in the morning Wiles announced his proof of Fermat's Last Theorem as a corollary to his main results. Having written the theorem on the blackboard he said I will stop here and sat down. In fact Wiles had proved the Shimura-Taniyama-Weil Conjecture for a class of examples, including those necessary to prove Fermat's Last Theorem.

    This, however, is not the end of the story. On 4 December 1993 Andrew Wiles made a statement in view of the speculation. He said that during the reviewing process a number of problems had emerged, most of which had been resolved. However one problem remains and Wiles essentially withdrew his claim to have a proof.

    In fact, from the beginning of 1994, Wiles began to collaborate with Richard Taylor in an attempt to fill the holes in the proof. However they decided that one of the key steps in the proof, using methods due to Flach, could not be made to work. They tried a new approach with a similar lack of success. In August 1994 Wiles addressed the International Congress of Mathematicians but was no nearer to solving the difficulties.

    Taylor suggested a last attempt to extend Flach's method in the way necessary and Wiles, although convinced it would not work, agreed mainly to enable him to convince Taylor that it could never work. Wiles worked on it for about two weeks, then suddenly inspiration struck.

    In a flash I saw that the thing that stopped it [the extension of Flach's method] working was something that would make another method I had tried previously work.

    On 6 October Wiles sent the new proof to three colleagues including Faltings. All liked the new proof which was essentially simpler than the earlier one. Faltings sent a simplification of part of the proof.

    No proof of the complexity of this can easily be guaranteed to be correct, so a very small doubt will remain for some time. However when Taylor lectured at the British Mathematical Colloquium in Edinburgh in April 1995 he gave the impression that no real doubts remained over Fermat's Last Theorem.

    Simon Singh wrote a book cunningly titled Fermats last theorem which is a great read for non mathematicians like myself, and having read it recently I thoroughly recommend it. It'll take you on a journey of discovery through mathematical history.
  8. 08 Sep '04 15:47 / 3 edits
    Originally posted by PTdissimulo
    Going back to Fermats theorem, the problem itself it really easy to understand, its the solution which is the problem

    For reference and because its not been included abouve in this thread the problem is this:

    Prove that xn + yn = ...[text shortened]... l take you on a journey of discovery through mathematical history.
    Very much agree re: Simon Singh's book. As was posted, it's entitled Fermat's Last Theorem and - while it does address issues directly relating to FLT and ultimately Wiles's work- is a roundabout meandering journey and exploration through many areas of interest. Terrifically entertaining stuff for the person with a passing (perhaps secretive!) interest in mathematics, logic and woman dressed up as men.

    The story of the friendship between the mathematicians Goro Shimura and Yutaka Taniyama is particularly touching.
  9. 09 Sep '04 10:05
    Originally posted by T1000
    Very much agree re: Simon Singh's book. As was posted, it's entitled Fermat's Last Theorem and - while it does address issues directly relating to FLT and ultimately Wiles's work- is a roundabout meandering journey and exploration through many areas of interest. Terrifically entertaining stuff for the person with a passing (perhaps secretive!) interest in ...[text shortened]... endship between the mathematicians Goro Shimura and Yutaka Taniyama is particularly touching.
    Thank you.
    I acually saw the book in a bookstore last friday, bought it
    and read it this weekend.
    Probably the best book I've ever read.
    Wonderful.
  10. Standard member genius
    Wayward Soul
    10 Sep '04 10:52
    Originally posted by TheMaster37
    What i know of it (and that isn't much) is that everal new branches of math has to be discovered before it could be proven. Furthermore it required to cooperation of alot of different theorems.Some of those were big. The complexity of the proof was so great that a single person couldn't get it to work, but several persons found bits and pieces, until someone realised that the last step in completing the proof was found.
    ahh-but you forget that fermeat himself claimed to have solved it! so, if i remeber correctly, there is still a prize on the go from the university of paris (paris? some random french one anyway...) for solving it using only maths that was around at the time of fermeat...