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    Hey Maths groobers!
    I never really got involved in your rarified world but this week's Horizon features this mathematician, Andrew Wiles who shut himself away for 7 years to prove that
    a^n+b^n does not equal c^n where n= interger>2

    Computers can't do it (yet).

    So, I wondered, what engineering or other use there is for this information?

    Could there be a better, more precise proof?

    And, what's the next challenge(s)?
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    (Original post by NJA)
    So, I wondered, what engineering or other use there is for this information?

    Could there be a better, more precise proof?

    And, what's the next challenge(s)?
    Most mathematical results don't have any use, but the methods used in their proof are often enlightening. As far as I'm aware most of the big "tools" used in the proof have little use (with the exception of Elliptic curve cryptography (my history isn't great, but I don't think ECs were studied because of FLT though)). But on the other hand I believe that FLT did inspire the entire development of Algebraic number theory amongst other topics.

    There could be, but I'm pretty doubtful. (For the record though, the proof is pretty precise).

    The Millenium prize problems include 6 of the biggest unsolved problems in mathematics, but there is hardly a shortage of unsolved problems in mathematics.
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    That's annoying when on earth did they start putting Horizon on BBC 4 I thought they had gone on a break!
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    Thanks for the link NJA.
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    (Original post by soup)
    That's annoying when on earth did they start putting Horizon on BBC 4 I thought they had gone on a break!
    It's also available on your TV if you have "catch up on demand", I think it was tuesday night.
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    (Original post by NJA)
    Hey Maths groobers!
    I never really got involved in your rarified world but this week's Horizon features this mathematician, Andrew Wiles who shut himself away for 7 years to prove that
    a^n+b^n does not equal c^n where n= interger>2

    Computers can't do it (yet).

    So, I wondered, what engineering or other use there is for this information?

    Could there be a better, more precise proof?

    And, what's the next challenge(s)?
    There is currently no use for the information, as far as I am aware. That said, proving this result had implications much more complicated than they appear across a number of other areas of maths. The theorem itself was proved by proving something that at a glance looks completely unrelated. However, in ten years or so, maybe there will be a use. Maths is literally years ahead of other subjects, in as much as that computer scientists and engineers invent new technology that relies on results that were produced by mathematicians years ago, and that appeared useless at the time. So really, it could be useful in any number of different ways.

    What do you mean by better, or more precise? Wiles's proof proves it, can you get more precise than that?

    Maths is full of challenges. Given ten minutes on the internet you could find a hundred unproven number theory theorems, and given a day you could find arbitrarily many routes to look down, if you knew where to look. The biggest problem currently in mathematics is probably the Riemann hypothesis, which can't be expressed in quite as simple terms as Fermat's theorem, but there are any number of other tremendously significant problems waiting to be solved.
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    (Original post by NJA)

    Computers can't do it (yet).
    I don't think there's the need for "(yet)"
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    (Original post by NJA)
    Computers can't do it (yet).
    or ever

    edit: beaten by daniel
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    Oh yeah, I came up with a neat proof for that once, but it can't fit in this text box.

    ...

    I think the proof of this problem has garnered so much attention because of its simplicity. I don't think there's much point in the solving of it - it's just a famous problem. But having said that, I've only read 1 or 2 books on the subject, and I was half asleep during them (not because they weren't interesting) - I don't really think I'm qualified to talk about it at length.
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    (Original post by Bobifier)
    ...What do you mean by better, or more precise? Wiles's proof proves it, can you get more precise than that?
    Chess masters like to gain a position in the smallest number of moves.

    I'm told that some computer languages are better than others.

    If I had a better command of the English language I could present my points in a way that others could more easily understand - same principle.
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    (Original post by NJA)
    Chess masters like to gain a position in the smallest number of moves.

    I'm told that some computer languages are better than others.

    If I had a better command of the English language I could present my points in a way that others could more easily understand - same principle.
    Maths isn't chess. Furthermore, chess masters go for the position that is most advantageous, not the one that is fastest.

    Yes, they are, and that equally has no bearing on how a proof can be better. They are two unrelated things.

    In maths, once something is proven, most people are happy with that. It is often possible to create a shorter and more elegant proof, but to say that it is a better proof would be pushing it. Both proofs serve the same purpose equally well.
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    (Original post by Bobifier)
    It is often possible to create a shorter and more elegant proof, but to say that it is a better proof would be pushing it. Both proofs serve the same purpose equally well.
    Unless that's how we define a 'better' proof?

    But there are other natural definitions of 'better' in this sense, like the proof which gives you the 'why', not just the 'oh look, i proved it'.
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    (Original post by Pheylan)
    or ever

    edit: beaten by daniel
    There's good reason to believe this, but there's no evidence. For instance, if it can be shown that every formal sentence (in first-order logic, say) is has a proof which has polynomial length, and if a polynomial-time algorithm is invented to find such proofs, we would have a constructive (!) proof that P = NP. (Actually, it would be even stronger than that, because it suffices to solve the Boolean satisfiability problem, which is equivalent to finding disproofs of propositions.) But we tend to believe that NP is strictly bigger than P, so we should also believe there is either no such algorithm, or that lengths of proofs can grow faster than polynomials.
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    (Original post by Zhen Lin)
    There's good reason to believe this, but there's no evidence. For instance, if it can be shown that every formal sentence (in first-order logic, say) is has a proof which has polynomial length, and if a polynomial-time algorithm is invented to find such proofs, we would have a constructive (!) proof that P = NP. (Actually, it would be even stronger than that, because it suffices to solve the Boolean satisfiability problem, which is equivalent to finding disproofs of propositions.) But we tend to believe that NP is strictly bigger than P, so we should also believe there is either no such algorithm, or that lengths of proofs can grow faster than polynomials.

    Kurt Gödel proved that for any set of axioms, there are theorems that exist that are not provable (they cannot be proved to be true or false). So that idea is a dead end.
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    (Original post by in_jeopardy)
    Kurt Gödel proved that for any set of axioms, there are theorems that exist that are not provable (they cannot be proved to be true or false). So that idea is a dead end.
    I'm afraid that's not quite what Gödel proved. Gödel's incompleteness theorem is about theories (logical systems) which have enough power to talk about proofs within themselves, in particular, any system with arithmetic. Moreover, there's a certain subtlety to it. The theorem tells us, for instance, that there is no proof within Peano arithmetic that tells us that Peano arithmetic is consistent. But there's a trivial proof in Zermelo-Fraenkel set theory that Peano arithmetic is consistent: \omega = \mathbb{N} is certainly a legal set, and it (together with certain arithmetic operations to be defined) satisfies all the axioms of Peano arithmetic, so it is a model, and a first-order theory has a model if and only if it is consistent. But then the theorem tells us that there is no proof in set theory that set theory is consistent.

    There are plenty of theories which are complete in the sense that every sentence either has a proof or a disproof. For instance, the theory of free \mathbb{Z}-sets is complete - that is to say, if you have a set X and a bijective function f: X \to X such that \forall x [ f^n(x) = f^m(x) \iff n = m ], then any well-formed formal sentence whatsoever you can make using f, variable symbols, quantifiers, and logical connectives can be either proven or disproven using nothing more than the properties of f.
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    (Original post by soup)
    That's annoying when on earth did they start putting Horizon on BBC 4 I thought they had gone on a break!
    This Horizon was the one from ages ago written by Simon Singh. It's not even in 16:9 ratio!
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    What I'm more interested in is what was Fermat's original proof of his own theorem that has eluded us all these years, as by Andrew Wiles' own accord his proof is a 21st century one and could not possibly be the same as that of Fermat's.

    Also, Simon Singh for all his academic calibre, needs to sort his bloody hair out.
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    (Original post by TheCrackFox)
    What I'm more interested in is what was Fermat's original proof of his own theorem that has eluded us all these years, as by Andrew Wiles' own accord his proof is a 21st century one and could not possibly be the same as that of Fermat's.
    Well the question arising is; did Fermat have a solution because these guys with their brilliant minds can make theorems with out proving them especially this one since it maybe considered really simple compared to other mathematical theorems (compare the statements and you will see the difference )
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    (Original post by umbrella3000)
    This Horizon was the one from ages ago written by Simon Singh. It's not even in 16:9 ratio!
    Yeah I realised that when I saw the end credits! I'm such an idiot
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    (Original post by TheCrackFox)
    What I'm more interested in is what was Fermat's original proof of his own theorem that has eluded us all these years, as by Andrew Wiles' own accord his proof is a 21st century one and could not possibly be the same as that of Fermat's.

    Also, Simon Singh for all his academic calibre, needs to sort his bloody hair out.
    If I'm remembering correctly, nearer the end of his life, Fermat published proofs for some small cases(n=3,4,5 or something), suggesting that either his original proof had a mistake, or hadn't existed. But I may be wrong.
 
 
 
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