It’s been proven that you can’t prove this … without some knowledge of basic math.
Yes, of course it is, as we’ve been over many, many times before. (Assuming you take the standard real numbers, as opposed to some other numbers with entirely different rules.) Wikipedia has an entire article on this, complete with proofs.
To further shit on his parade, no government has ever offered a prize for squaring the circle:
‘Cyclometer’ is an obsolete term for someone who intends to square the circle. (And, of course, the longitude problem does not depend on squaring the circle. We know this because the longitude problem was actually solved.)
George Dantzig famously solved two previously unsolved problems under the mistaken belief that they were homework.
I think I’d consider the Collatz problem to be either a silly recreational math problem or the start of something like a theory of dynamic systems or ergodic theory for Z. A lot of combinatorics is that way: it’s a mixture of ad hoc methods for somewhat arbitrary and uninteresting problems, plus some genuinely deep stuff.
Aside from Fermat’s Last Theorem’s being a famously unsolved (for a long time, at least) problem, its mathematical significance is not so much for its own merits as from the fact it’s a corollary of the Taniyama–Shimura conjecture, which is extremely important. (Proving that it follows from the conjecture is nontrivial.) This is Serious Business in the subject, while FLT itself is interesting but not especially significant.
What probably happened with Fermat, meanwhile, is that he did in fact find a method which worked to prove the n=3 and n=4 cases (which he later published), and mistakenly believed that the method would generalize to arbitrary n. When he discovered that it didn’t, he never bothered to write a retraction of his hasty marginal note, because after all, it was only a hasty marginal note. It’s not like it was published.
What’s been proven (right here on the SDMB!) is that no finite amount of proof suffices to prove it to some people. 
I’m going to clarify my wording here in a very pedantic way which likely no one else will much care about:
Indeed, we already know, for every consistent formal proof system T, that if there’s any polytime program T proves to solve NP-complete problems, then there’s a very particular polytime program solving NP-complete problems which we already know how to write [the “Run, in dovetailed parallel, all programs which T thinks solves this problem” program]. Though this program will not itself be proven correct by T, per se; rather, its correctness will be equivalent to the consistency of T (which is unprovable in T for Goedelian reasons).
Perhaps not. If P != NP is true it’s likely to be unprovable! Here is a brief paper which points to the famous Razborov-Rudich paper which may suggest this fact.
Well, the Razborov-Rudich result indicates only that a certain, very particular strategy for proofs popular throughout complexity theory cannot be used to establish P != NP (and even this result depends on certain further unproven complexity-theoretic assumptions).
This is a noteworthy result, but it’s far from establishing (even under the assumptions of the result) the unprovability of P != NP by other commonly accepted means; it’s just the ruling out of one particular tactic. Even the article you link to says “the Razborov–Rudich result should be regarded as a hint, and not a barrier, to separating complexity classes. The only real barrier is our lack of imagination.”
Whoosh
robert_columbia writes:
> George Dantzig famously solved two previously unsolved problems under the
> mistaken belief that they were homework.
O.K., but let’s be clear about Dantzig’s status. He wasn’t someone with no previous work in mathematics who one day saw a problem and solved it immediately. He had already gotten a master’s degree in math and worked for a couple of years as a statistician. He was a grad student at Berkley working on his Ph.D. He saw the problem which he assumed was just a homework assignment and took several days to solve it.
The Razborov-Rudich result establishes a particular case of a much more general intuitive argument. I won’t expand on this: Dopers would only debate whether to request a cite on the intuition or to charge the messenger with having none. :dubious:
Well, one reason I commented was to highlight, as always, that there’s no such thing as “unprovable” simpliciter; only “unprovable in this system”, “unprovable in that system”, etc. Without qualification, “unprovable” is meaningless.
The canonical case of this is the Dot Conjecture.
Hmm… I thought I’d read that it was believed that he had solved it except for the problem of unique factorisation. Or, he forgot to explicitly limit his solution to cardinals when making his margin note. It was, after all, only a margin note.
What “problem of unique factorization”? That’s the Fundamental Theorem of Arithmetic, and is widely known to everyone doing anything in number theory.
While each cardinal has a unique factorisation of cardinals (e.g. 12 = 2x3x3), this is not the case if you allow complex numbers as factors - 12 = (1+ SQRT(-11)) x (1-SQRT(-11)). This issue was mostly, but unfortunately not completely, fixable.
See Simon Singh’s Fermat’s Last Theorem, pp120-128.
It was Gabriel Lamé, not Fermat, who thought he had proven Fermat’s Last Theorem, but had in fact mistakenly assumed that the ring generated by a pth primitive root of unity, for any fixed prime p, forms a unique factorization domain.
And Cauchy. But the hypothesis is that Fermat also hit on unique factorisation but forgot about complex numbers.
I remember hearing about the four color map theorem when I was about seven. I immediately started drawing maps to try to disprove it. Without success. :rolleyes:
Ah, right, I’d forgotten that.
I suppose that’s possible. I’d never heard that before; do you know what evidence there is for it?