# Which famous mathematical puzzles have not been solved yet? **URL:** \ **Category:** Factual Questions\ **Created:** [September 6, 2013, 5:39am UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152 "2013-09-06T05:39:35Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [September 6, 2013, 6:41pm UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/21 "2013-09-06T18:41:53Z")
> [@Asympotically\_fat](#): > > Does 0.999…=1? Nobody knows:confused: It’s been proven that you can’t prove this … without some knowledge of basic math. ---
**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\ **Post date:** [September 6, 2013, 10:34pm UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/22 "2013-09-06T22:34:00Z")
> [@Asympotically\_fat](#): > > Does 0.999…=1? Nobody knows:confused: 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.](http://en.wikipedia.org/wiki/0.999...) > [@Leo\_Bloom](#): > > Leopold Bloom thinks about how he may afford to pay for a country residence: > > **What rapid but insecure means to opulence might facilitate immediate purchase?** > […] A solution of the secular problem of the quadrature of the circle, government premium £1,000,000 sterling. [To further shit on his parade, no government has ever offered a prize for squaring the circle](http://en.wikipedia.org/wiki/Squaring_the_circle#Connection_with_the_longitude_problem): > [@Augustus de Morgan](#): > > Montucla says, speaking of France, that he finds three notions prevalent among cyclometers: 1. That there is a large reward offered for success; 2. That the longitude problem depends on that success; 3. That the solution is the great end and object of geometry. The same three notions are equally prevalent among the same class in England. No reward has ever been offered by the government of either country. ‘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.) ---
**Author:** ![robert\_columbia](https://avatars.discourse-cdn.com/v4/letter/r/e79b87/32.png) [@robert\_columbia](https://boards.straightdope.com/u/robert_columbia)\ **Post date:** [September 6, 2013, 10:49pm UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/23 "2013-09-06T22:49:13Z")
> [@Wendell\_Wagner](#): > > Given that you’ve called them “puzzles” rather than “problems,” I wonder if you think that they’re cute little puzzles that you might get lucky and solve in a long afternoon. They’re not. They’re the sort of problem that you’d generally need to get your Ph.D. in math and then spent several years of your free time working on to have a good chance at solving them. Often it takes a considerable background in math to be able to understand what the question is. > [@Thudlow\_Boink](#): > > Although some of them give that impression. To the naïve, things like the Goldbach Conjecture or the Collatz Conjecture _look_ sort of like cute little puzzles that one _ought_ to be able to figure out just by thinking about them hard enough. (Fermat’s Last Theorem, too, if Fermat had been correct about having a valid proof… George Dantzig famously [solved](http://www.umass.edu/wsp/statistics/tales/dantzig.html) two previously unsolved problems under the mistaken belief that they were homework. ---
**Author:** ![Itself](https://avatars.discourse-cdn.com/v4/letter/i/d07c76/32.png) [@Itself](https://boards.straightdope.com/u/Itself)\ **Post date:** [September 6, 2013, 11:17pm UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/24 "2013-09-06T23:17:08Z")
> [@Thudlow\_Boink](#): > > To the naïve, things like the Goldbach Conjecture or the Collatz Conjecture _look_ sort of like cute little puzzles that one _ought_ to be able to figure out just by thinking about them hard enough. (Fermat’s Last Theorem, too, if Fermat had been correct about having a valid proof that wouldn’t fit in the margin.) 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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 7, 2013, 12:06am UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/25 "2013-09-07T00:06:54Z")
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. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [September 7, 2013, 5:46am UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/26 "2013-09-07T05:46:45Z")
> [@Exapno\_Mapcase](#): > > It’s been proven that you can’t prove this … without some knowledge of basic math. What’s been proven (right here on the SDMB!) is that no finite amount of proof suffices to prove it to some people. 😛 ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 7, 2013, 8:10am UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/27 "2013-09-07T08:10:33Z")
> [@Indistinguishable](#): > > (Indeed, we already know, for every formal proof system T, that if there’s any program T proves to solve NP-complete problems in polytime, then there’s a very particular such program which we already know how to write [the “Run, in dovetailed parallel, all programs which T thinks solves this problem” program]). 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). ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [September 7, 2013, 8:18am UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/28 "2013-09-07T08:18:30Z")
> [@Jragon](#): > > Of course, it’s far more likely that some professor or industrial researcher will someday prove P =/= NP with a boring 20 page proof incomprehensible … 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](http://www.ams.org/notices/201111/rtx111101586p.pdf). ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 7, 2013, 8:46am UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/29 "2013-09-07T08:46:14Z")
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.” ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [September 7, 2013, 8:49am UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/30 "2013-09-07T08:49:07Z")
> [@Derleth](#): > > 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.](http://en.wikipedia.org/wiki/0.999...) Whoosh ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [September 7, 2013, 9:58am UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/31 "2013-09-07T09:58:17Z")
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. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [September 7, 2013, 10:14am UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/32 "2013-09-07T10:14:34Z")
> [@Indistinguishable](#): > > 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)… > The only real barrier is our lack of imagination." 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: ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 7, 2013, 10:28am UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/33 "2013-09-07T10:28:59Z")
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. ---
**Author:** ![Quartz](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/quartz/32/267_2.png) [@Quartz](https://boards.straightdope.com/u/Quartz)\ **Post date:** [September 7, 2013, 1:28pm UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/34 "2013-09-07T13:28:25Z")
> [@Jragon](#): > > but there are certainly hard problems that could, conceivably, be solved accidentally in an afternoon with absolutely no formal understanding of the problem at hand. The canonical case of this is the Dot Conjecture. > [@Chronos](#): > > 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. 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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 7, 2013, 2:38pm UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/35 "2013-09-07T14:38:29Z")
What “problem of unique factorization”? That’s the Fundamental Theorem of Arithmetic, and is widely known to everyone doing anything in number theory. ---
**Author:** ![Quartz](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/quartz/32/267_2.png) [@Quartz](https://boards.straightdope.com/u/Quartz)\ **Post date:** [September 7, 2013, 5:30pm UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/36 "2013-09-07T17:30:51Z")
> [@Chronos](#): > > What “problem of unique factorization”? 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. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 7, 2013, 5:52pm UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/37 "2013-09-07T17:52:59Z")
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. ---
**Author:** ![Quartz](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/quartz/32/267_2.png) [@Quartz](https://boards.straightdope.com/u/Quartz)\ **Post date:** [September 7, 2013, 7:09pm UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/38 "2013-09-07T19:09:31Z")
> [@Indistinguishable](#): > > It was Gabriel Lamé, not Fermat, who thought he had proven Fermat’s Last Theorem, And Cauchy. But the hypothesis is that Fermat also hit on unique factorisation but forgot about complex numbers. ---
**Author:** ![standingwave](https://avatars.discourse-cdn.com/v4/letter/s/9de0a6/32.png) [@standingwave](https://boards.straightdope.com/u/standingwave)\ **Post date:** [September 7, 2013, 7:30pm UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/39 "2013-09-07T19:30:32Z")
I remember hearing about the [four color map theorem](http://en.wikipedia.org/wiki/Four_color_theorem#False_disproofs) when I was about seven. I immediately started drawing maps to try to disprove it. Without success. :rolleyes: ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 7, 2013, 7:49pm UTC](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152/40 "2013-09-07T19:49:32Z")
> [@Quartz](#): > > And Cauchy. Ah, right, I’d forgotten that. > [@Quartz](#): > > But the hypothesis is that Fermat also hit on unique factorisation but forgot about complex numbers. I suppose that’s possible. I’d never heard that before; do you know what evidence there is for it? [Previous page](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152.md?page=1) [Next page](https://boards.straightdope.com/t/which-famous-mathematical-puzzles-have-not-been-solved-yet/668152.md?page=3)