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Fermat's last theorum

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Reply 20
Avatar for Zhen Lin
Zhen Lin
12
Original post by Slumpy
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.


He published a proof for the case n = 4, which was an argument by infinite descent. I think he also published an incorrect proof of the case n = 3 — in any case, Euler published a correct proof later. It is speculated that, yes, Fermat realised his original argument was flawed somehow.
Reply 21
Avatar for Slumpy
Slumpy
16
Original post by Zhen Lin
He published a proof for the case n = 4, which was an argument by infinite descent. I think he also published an incorrect proof of the case n = 3 — in any case, Euler published a correct proof later. It is speculated that, yes, Fermat realised his original argument was flawed somehow.


I knew it was 2 of 3,4,5, and thought one was wrong, but no more than that. Clearly need to brush up on my maths history a tad!
Reply 22
Avatar for Kolya
Kolya
17
Original post by Zhen Lin

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.


Sorry, why is there good reason to believe a computer could never prove FLT? Automated theorem proving already exists. Proving something like the FLT would be a huge step-up, but I don't see why it would be impossible.
Reply 23
Avatar for Pheylan
Pheylan
i thought that by this

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).


NJA meant that computers aren't yet able to provide integer solutions to a^n+b^n=c^n for n>2, but they might be able to in the future

probably should've read the post properly :getmecoat:
Original post by Kolya
Sorry, why is there good reason to believe a computer could never prove FLT? Automated theorem proving already exists. Proving something like the FLT would be a huge step-up, but I don't see why it would be impossible.


Indeed, computers have proved FLT, they happen to be biological, but there is nothing particular about human brains that means they can't be simulated by some sufficiently advanced artificial computer.

My knowledge of complexity theory amounts to a few sound-bites, but I think the implication of P≠NPP \neq NP being alluded to here is that there even though proofs may be polynomial time checkable, there is no general polynomial time algorithm for generating them.

This doesn't mean that computers are theoretically forbidden from finding difficult proofs, it just means that it will not be a mechanised polynomial time process . You would rely on heuristics, guess work some brute force, and the like- much as humans do. Which is nice because it means computers need to be as creative as people.
Reply 25
Avatar for RichE
RichE
15
Original post by Pheylan

NJA meant that computers aren't yet able to provide integer solutions to a^n+b^n=c^n for n>2, but they might be able to in the future


How? We know there aren't any :confused:
(edited 13 years ago)

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