107

What is the oldest open problem in mathematics? By old, I am referring to the date the problem was stated.

Browsing Wikipedia list of open problems, it seems that the Goldbach conjecture (1742, every even integer greater than 2 is the sum of two primes) is a good candidate.

The Kepler conjecture about sphere packing is from 1611 but I think this is finally solved (anybody confirms?). There may still be some open problem stated at that time on the same subject, that is not solved. Also there are problems about cuboids that Euler may have stated and are not yet solved, but I am not sure about that.

A related question: can we say that we have solved all problems handed down by the mathematicians from Antiquity?

Martin Sleziak
  • 4,608
coudy
  • 18,537
  • 5
  • 74
  • 134
  • 49
    Existence of odd perfect numbers? – Andrey Rekalo Jun 04 '10 at 18:54
  • 55
    Since this is a question about a point-of-fact rather than a big-list question, people should not post new answers unless they also provide an argument that their proposal predates all previously given answers. – Noah Snyder Jun 04 '10 at 18:55
  • 9
    The Kepler Conjecture has been solved, but there is some controversy since the proof makes extensive use of computers. There is a project underway to produce a formal proof called Project Flyspeck.

    http://code.google.com/p/flyspeck/

     – Tony Huynh Jun 04 '10 at 19:20
  • 6
    As Noah hints at in his answer, I imagine you can throw a rock into the ocean and hit an unsolved number theory question from ancient Greece. – Gunnar Þór Magnússon Jun 04 '10 at 19:23
  • 6
    @Gunnar. I am not asking if the Greeks could have asked some mathematical problem that we can't solve today. I am asking if they actually asked such a problem, which is very different (see my comment to Noah's answer). – coudy Jun 04 '10 at 19:42
  • 1
    Thomas Hales has proved Kepler's conjecture, but it uses enough computer calculation that there are still holdouts. He is currently working on producing a computer-verified proof. – Kevin O'Bryant Jun 04 '10 at 21:16
  • What is a perfect strategy in chess would be a good contender. – TROLLHUNTER Oct 20 '13 at 10:38
  • Wasn't the concept of the very existence of a perfect chess strategy only as old as the respective paper by Zermelo? (This would mean that the question about chess was not a good contender after all). – Włodzimierz Holsztyński Apr 11 '14 at 07:47
  • 1
    I'm having trouble believing that the answer to this question is anything other than the twin prime conjecture. I think just about everyone who looks at a list of the first 50 primes formulates this conjecture, even if he or she does not make a serious attempt to prove it. – Paul Siegel May 09 '14 at 15:11
  • 1
    The Flyspeck project, mentioned above, was completed in 2014. – Timothy Chow Nov 26 '22 at 18:09

10 Answers10

89

Existence or nonexistence of odd perfect numbers.

Update: Goes back at least to Nicomachus of Gerasa around 100 AD, according to J J O'Connor and E F Robertson. Nichomachus also asked about infinitude of perfect numbers.

(Goes back at least to Descartes 1638 https://mathworld.wolfram.com/OddPerfectNumber.html and arguably all the way back to Euclid.)

Martin Sleziak
  • 4,608
Noah Snyder
  • 27,820
  • 1
    Reading the link you provide about Nicomachus, it looks to me as if he stated that all perfect numbers are even. So he didn't regard it as an open problem. – TonyK Jun 04 '10 at 19:58
  • 12
    Surely you would say that Fermat's Last Theorem (before it was solved) was an open problem dating back to Fermat! – Noah Snyder Jun 04 '10 at 20:03
  • 1
    @Noah. Thanks for the link. It definitely answers my second question. – coudy Jun 04 '10 at 20:03
  • 10
    Should we credit someone with posing X as an open problem if he made a claim implying that X is true? That's debatable. Dickson, History of theory of numbers, vol 1 says that Nicomachus classified $\textit{even}$ numbers into abundant, deficient and perfect and that he claimed that every perfect number is obtained by Euclid's rule. However, it's not clear that Nicomachus knew of existence of odd abundant numbers! So while we may speculate that Nicomachus has considered the question, as far as I can tell, there is no indication of it in his book. That makes the situation different from FLT. – Victor Protsak Jun 04 '10 at 21:04
  • Stretching it a bit further, we may ascribe the problem of finding perfect numbers to Euclid. He proved a formula for some of them, which is indirect evidence that he wanted to know them all. For the record, I disagree with this argument, but it's akin to crediting Nicomachus with the odd perfect number problem. – Victor Protsak Jun 04 '10 at 21:12
  • 8
    So the reason I'm slightly uncomfortable going back all the way to Euclid is that "find all perfect numbers" is too imprecise a question to be called an open problem. Otherwise we could say "find all prime numbers" is a problem going back to whenever people started looking for them. "All perfect numbers are odd" or "there are infinitely many perfect numbers" are both well-formed specific problems. And certainly they were problems raised by Nicomachus as you can't (bogusly) answer a question without first having a question! – Noah Snyder Jun 04 '10 at 22:17
  • An analogous situation with primes would be to give a $\textit{formula}$ for some primes, not just to consider primes. I don't see why "find all perfect numbers" would be illegitimate as a specific problem, $\textit{had it been explicitly asked}.$ After all, no one objects to "find all even perfect numbers" being a well-formed specific question (even though its answer involves Mersenne primes, which we don't completely know). Just to emphasize: I don't think we should go all the way back to Euclid, or to Pythagoreans, for that matter, because they already knew 6 and 28 were perfect numbers. – Victor Protsak Jun 04 '10 at 22:43
  • 1
    What I disagree with is that the problem of non-existence of odd perfect numbers was $\textit{raised}$ by Nicomachus: I don't see any evidence for that. For all we know he could have simply overlooked the possibility. There have been many wrong claims made in history of mathematics, including the perfect numbers problem, as enumerated by Dickson. Should we interpret each one of them as an implicit question? That would be rather unorthodox. – Victor Protsak Jun 04 '10 at 22:44
  • 1
    I feel like we're talking past each other. Nichomachus did explicitly ask these questions when he claimed to answer them! But on the specific point I don't think "find all even perfect numbers" is specific enough to be a "problem" (as opposed to a subject of inquiry), but I do think "does Euclid's formula give all even perfect numbers" is a specific problem. However, it's not clear from Euclid that he ever thought that formula gave all even perfect numbers, so I'm not comfortable crediting him with that problem. – Noah Snyder Jun 04 '10 at 22:49
  • 5
    Also I think there's a difference between "crediting X with posing Y as an open problem (when they really said they'd answered Y)" and simply saying "Y is an open problem which dates back at least to the work of X (who falsely claimed to have solved it)." They're different in two ways, first "credit" is too positive a word for this situation, and second "posing X as an open problem" is only one of way that a problem can first appear in the literature. – Noah Snyder Jun 04 '10 at 22:53
  • 1
    Yes, we are talking past other, I fear! When I say "P overlooked X", I truly mean "P didn't even consider the possibility of X", which seems to be the case here (only even numbers categorized as abundant, perfect, or efficient). Forget "credit", that may be a poor word choice. Suppose that P made a claim Z that, logically speaking, implies X. I agree that "X dates back to the work of P", but unless there is evidence that P considered X itself, and not Z, I wouldn't express the situation as "P falsely claimed to have solved X" or that "X appeared in the work of P". – Victor Protsak Jun 05 '10 at 00:07
  • So, you're claiming that the article I linked to is wrong, the excerpt you give from Dickson doesn't seem enough to draw that conclusion. I tried to find the original source on google books and it doesn't appear to be there. – Noah Snyder Jun 05 '10 at 00:09
  • At best, you can say that Nicomachus claimed that Euclid's rule gives all perfect numbers, which implies that (a) all even perfect numbers arise in this way (true, proved by Euler) and (b) there are no odd numbers (still unknown). What $\textit{I}$ am uncomfortable with is basing a categorization of X as a "problem" or "subject of inquiry" on $\textit{a posteriori}$ existence of a clean answer to it. – Victor Protsak Jun 05 '10 at 00:18
  • 1
    Actually, the McTutor article does quote the passage from which they deduce "assertion (2)" that all perfect numbers are even. My take on it is that $\textit{particular numbers generated by Euclid's algorithm [described right before] are all even},$ which is true. The authors seem to say that there is nothing in that passage beyond illustrating Euclid's rule by the first four perfect numbers 6,28,496,8128. You can write to them and ask whether they had any other evidence. As it is, their quote does not support their attribution of assertion (2) to Nicomachus, in my opinion. – Victor Protsak Jun 05 '10 at 00:52
  • No, that's just the excerpt of point (4). They didn't excerpt the statements of the other points. – Noah Snyder Jun 05 '10 at 01:07
  • Right after "Some of the assertions are made in this quote about perfect numbers which follows the description of the algorithm". – Victor Protsak Jun 05 '10 at 01:46
  • 1
    Ah, ok, you seem to be right that it's more accurate to say that the claim here is that Euclid's description gives all perfect numbers. Still, that's essentially the same open question. – Noah Snyder Jun 05 '10 at 01:55
  • 23
    @TonyK: I bet Nicomachus had a proof, probably a very remarkable one, but the margin of his papyrus... well, you know how it is. – Nate Eldredge Jun 05 '10 at 03:42
  • 1
    @NoahSnyder: In the Wikipedia section on odd perfect numbers, it is stated that "In 1496, Jacques Lefèvre stated that Euclid's rule gives all perfect numbers, thus implying that no odd perfect number exists." The cited reference is Dickson's History of the Theory of Numbers, Vol. I (1919), p. 6. – Jose Arnaldo Bebita Dris Nov 29 '21 at 08:19
44

The Congruent Number problem (Which integers are the areas of right triangles with rational sides?) dates back to an Arab manuscript written before 972 AD, according to https://www.jstor.org/stable/2320381.

Martin Sleziak
  • 4,608
TonyK
  • 2,191
  • 15
  • 15
  • Not as old or well-defined, but the systematic construction of quasicrystals appeared to be considered around 1200 CE: http://peterlu.org/content/decagonal-and-quasi-crystalline-tilings-medieval-islamic-architecture – Steve Huntsman Jun 04 '10 at 19:45
  • I have not read the reference, but is it true that no currently known (unconditional) general algorithm (allowed to be slow) can determine if an integer $n$ is congruent or not? Can you mention a particular $n$ for which the status (congruent or not) is unknown? – Jeppe Stig Nielsen Apr 29 '16 at 15:10
  • 2
    @JeppeStigNielsen: If you read the relevant Wikipedia article, then you will know as much as I do about congruent numbers :-) – TonyK Apr 29 '16 at 15:26
30

Another unsolved problem from ancient Greek times is: which regular $n$-gons are constructible by ruler and compass? We know, since Gauss, that this problem reduces to finding all the Fermat primes, but we don't know that we have found them all yet.

John Stillwell
  • 12,258
  • Is there evidence that the Greeks asked this question? – Gerald Edgar Jun 04 '10 at 22:44
  • 3
    Do you know whether the question was explicitly asked, outside of the context of trisecting the angle? I thought that a revolutionary aspect of Gauss's discovery was that it had been assumed no cases beyond classically known were possible, but I don't know whether there was a specific claim made to that effect. – Victor Protsak Jun 04 '10 at 22:53
  • 2
    Why wasn't constructing the 7-gon a famous question like squaring the circle, doubling the cube, and trisecting the angle? If the Greeks did consider this an open problem, why was it considered less important than the others? (Or is the emphasis of those three problems something that happened later?) – Noah Snyder Jun 04 '10 at 23:00
  • 22
    I can't say that the Greeks explicitly asked the question about $n$-gons, but they considered enough special cases that the question probably crossed their minds. Euclid has $n=3,4,5,6,15$ and there were attempts for $n=7$. Archimedes gave a construction of the 7-gon using "neusis" (a sliding ruler device that also allows trisection of angles) so the 7-gon problem was certainly of interest to the Greeks. – John Stillwell Jun 05 '10 at 01:33
  • "the question probably crossed their minds" ... the OP will have to tell us if that qualifies. – Gerald Edgar Jun 05 '10 at 03:37
  • @Gerald. Not really. I insist on providing a reference and a date, so that anybody can make its own opinion. Also the question about the seven-gon is solved, isn't it ?

    @John. Is it plausible that the greek actually asked "which regular n-gons are constructible by ruler and compass" ? It is more likely that they asked "Whether there are n-gons that can't be constructed with ruler and compass" (which is solved). The former question does not really make sense before you have the answer for the later. Any reference welcome.

     – coudy Jun 05 '10 at 07:38
  • 14
    @coudy: I think it very plausible that the Greeks asked whether there are $n$-gons that can't be constructed with ruler and compass, and that they thought $n=7$ is an example (because they were willing to use neusis to construct it).However, under your reference/date conditions I'm afraid no question from ancient Greek times is going to qualify. There is no exact date for the works of Euclid or Archimedes, and we don't have the original manuscripts. – John Stillwell Jun 05 '10 at 08:05
  • Just because they found a neusis construction for the regular heptagon (which I don't think is undisputed), it does not follow that they thought it was not constructible by Euclidean methods. They may have meant the neusis solution to be a "stopgap" pending a Euclidean one. – Oscar Lanzi Dec 31 '22 at 23:54
28

Albrecht Dürer's conjecture states that every convex polytope has a non-overlapping edge unfolding (see here for the intro). This problem was raised in 1525, revived by Shephard in 1975, and remains wide open.

Igor Pak
  • 16,290
  • 6
    I would like to supplement Igor's post: There seems to be no evidence that Dürer recognized this as a problem that needed a proof. I believe the problem was first posed as a clear problem in need of resolution in the Shephard paper Igor cited. So this runs into the same issue discussed by Noah and Victor et al. in the comments to Noah's accepted answer. – Joseph O'Rourke Jul 07 '10 at 01:27
14

Not exactly what you are asking for, but a candidate for the longest time elapsing between the proposal and the solution of a problem: the Archimedes cattle problem, proposed by Archimedes and solved by A. Amthor in 1880. See https://en.wikipedia.org/wiki/Archimedes%27s_cattle_problem

Martin Sleziak
  • 4,608
Richard Stanley
  • 49,238
  • 10
    It is dubious to attribute this to Archimedes--- it has the flavor of European 17-19th century puzzle mathematics, and it's attribution to Archimedes is folklore. – Ron Maimon Aug 01 '11 at 13:09
  • 3
    I retract my answer. – Richard Stanley Jul 06 '22 at 15:35
  • 1
    As per my answer (and others) here, modern classical scholarship seems to be that the problem is essentially authentic — that certainly some version of the problem genuinely goes back to Archimedes, and the specific poetic text of it we have today may be by Archimedes himself or, if not, was probably based on earlier versions that descended from him. – Peter LeFanu Lumsdaine May 28 '23 at 12:51
  • 1
    I retract my retraction. – Richard Stanley Oct 27 '23 at 17:56
10

According to Encyclopaedia Britannica, "Greek mathematician Euclid (flourished c. 300 bce) gave the oldest known proof that there exist an infinite number of primes, and he conjectured that there are an infinite number of twin primes," which makes the twin prime conjecture remarkably old.

Martin Sleziak
  • 4,608
Łukasz Grabowski
  • 3,887
  • 29
    "which would make the twin prime conjecture remarkably old". There's nothing on twin primes in the Elements, or, as far as I know, in any of Euclid's other writings that have survived. – Franz Lemmermeyer Oct 02 '10 at 17:53
  • 4
    quick googling strongly suggests that you're right. It's amazing how prevalent in internet (and in EB!) the contrary information is. – Łukasz Grabowski Oct 02 '10 at 18:25
  • 6
    Did EB confuse twin primes with perfect numbers? – Włodzimierz Holsztyński Apr 11 '14 at 07:43
  • 1
    Google seems to suggest the following "The first statement of the twin prime conjecture was given in 1846 by French mathematician Alphonse de Polignac, who wrote that any even number can be expressed in infinite ways as the difference between two consecutive primes" – Sidharth Ghoshal Jul 01 '23 at 06:19
8

This is not older than the rest, but old enough I believe: In 1775 Fagnano constructed periodic orbits for acutangular triangular billiards. The question about the existence of periodic orbits in general triangular (or polygonal) billiards (in the case of irrational angles) remains open. (

Troubetzkoy, Serge, Dual billiards, Fagnano orbits, and regular polygons, Am. Math. Mon. 116, No. 3, 251-260 (2009). arXiv:0704.0390 , jstor. ZBL1229.37033, MR2491981. ).

Martin Sleziak
  • 4,608
rpotrie
  • 3,878
  • Schlage-Puchta posted a paper to the arxiv recently with the title "On Triangular Billiards" which appears to provide the last piece in such a characterization. (I have concern about some things he states about Jacobsthal's function in that paper, but it doesn't appear to stop his main proof.) Gerhard "Ask Me About Jacobsthal's Function" Paseman, 2011.05.29 – Gerhard Paseman May 29 '11 at 15:47
  • Gerhard -- are you referring to http://arxiv.org/abs/1105.1629? This paper is on the classification for rational triangles, not on the existence of periodic trajectories in general (obtuse) triangles. – algori May 29 '11 at 22:12
  • Indeed I am referring to that paper. I apologize for my carelessness; I should have checked the literature with more care, and noticed the acute case specifically. Thank you for your clarification. Gerhard "Needs To Read Some More" Paseman, 2011.05.31 – Gerhard Paseman May 31 '11 at 07:16
3

The Perfect Cuboid problem was being considered in the early 18th century (according to https://mathworld.wolfram.com/EulerBrick.html )

I don't know if any ancient Greeks are on record as having considered the problem; but that doesn't seem beyond the bounds of possibility, although I would guess that in those times they were preoccupied mostly with 2-D problems.

Martin Sleziak
  • 4,608
John R Ramsden
  • 1,495
  • 2
    This is not a big-list question… – Dirk May 09 '14 at 16:05
  • 2
    'I would guess that in those times they were preoccupied mostly with 2-D problems.' See: http://en.wikipedia.org/wiki/Doubling_the_cube – HJRW May 10 '14 at 18:53
0

Zeno's paradoxes are among the oldest puzzles at the intersection of mathematics, philosophy, and physics (in alphabetical order). The traditional resolution of Zeno's paradoxes of motion involves modeling them in terms of the real line and interpreting the iterated procedure as an infinite series.

As pointed out in one of the comments, Heisenberg's uncertainty principle provides another way of accounting for the puzzle, by arguing that it has no physical meaning.

H. Jerome Keisler in his article "The hyperreal line" (207–237) in the collection

Real numbers, generalizations of the reals, and theories of continua. Edited by Philip Ehrlich. Synthese Library, 242. Kluwer Academic Publishers Group, Dordrecht, 1994

provides a different mathematical resolution of the puzzle in terms of the hyperreal continuum.

More recently (2013), Terry Tao notes the mathematical significance of these paradoxes by noting that they "make the important point that real analysis cannot be reduced to a branch of discrete mathematics, but requires additional tools in order to deal with the continuum" (see https://mathscinet.ams.org/mathscinet-getitem?mr=3026767).

In a review of Graham Oppy's book, John H. Mason makes the following intriguing comment, indicative of the richness of the issues involved: Have you ever briefly called upon Zeno's paradoxes when introducing the notion of limit to students? For example, the fact that Achilles really does catch the tortoise is only because he crosses distances halving in length in intervals of time also halving in length; the arrow does actually get to its target, even though it has to surmount an infinite number of decreasingly small intervals. This book addresses these and many other paradoxes involving infinitely large and infinitely small quantities with philosophical precision and reasoning. It reveals that there are much larger issues at stake than are perhaps commonly recognised, and certainly than are `dismissed' with the Cauchy-Weierstrass formalism of limits. See https://mathscinet.ams.org/mathscinet-getitem?mr=2238333

Martin Sleziak
  • 4,608
Mikhail Katz
  • 15,081
  • 1
  • 50
  • 119
  • 11
    OK, I'll bite. Surely this is resolved by the notion of convergence of infinite series? – HJRW May 08 '14 at 15:39
  • @HJRW, whether or not this has been resolved is certainly a matter of dispute; for a discussion see e.g., the wiki page I linked. – Mikhail Katz May 09 '14 at 07:19
  • 3
    The Wikipedia page does not suggest anywhere that the mathematical content of the paradox is unresolved. Any remaining problems seem to involve the question of how motion in the physical world should be described. Heisenberg's Uncertainty Principle strongly indicates that infinitely dividing the trajectory of a real-world moving object is not a meaningful thing to do. – S. Carnahan May 09 '14 at 08:35
  • 1
    @S.Carnahan, this is precisely my point. Since there are various ways to model mathematically the idea of motion, there is more than one possibility of accounting for the paradox. The uncertainty principle you mentioned is certainly a different way of accounting for it from the infinite series approach suggested by HJRW. The fact is that there are several recent mathematical papers providing yet a different account of the paradox. – Mikhail Katz May 09 '14 at 08:39
  • 11
    I tend to think that the possibility of various ways to model mathematically the idea of motion make it an open problem of physics rather than mathematics. – Emil Jeřábek May 09 '14 at 09:57
  • 1
    @Emil, this is a problem of physics only to the extent that the trajectory of a rock thrown in the air is a problem of physics. Certainly the rock is a problem of both mathematics and physics. Furthermore, the Zeno paradoxes have a theoretical aspect about them (infinitely many steps, etc.) that would place it outside of the realm of physics as ordinarily conceived. – Mikhail Katz May 10 '14 at 18:26
  • 1
    @S.Carnahan, I don't think the mathematics can be separated from physics in this case. Furthermore, the view that Zeno's paradoxes have a mathematical aspect to them is a fairly common view in the literature. – Mikhail Katz May 10 '14 at 18:29
  • Re rock: it splits into a physics problem to determine an adequate mathematical model of the physical reality in question (Newton’s laws, gravity, friction forces, ...), and a mathematical problem to figure out the trajectory in said model. Likewise for Zeno’s paradoxes, finding a mathematical model of the situation is a problem of physics, and working out the solution of the paradox in such a model is a problem of mathematics. Each model yields a different mathematical problem. In the recent literature you mentionn, if someone proposes a novel model of motion and immediately solves ... – Emil Jeřábek May 11 '14 at 14:17
  • ... the paradox, this does not give an mathematical problem (which in any case would be new). It only shows that the physics problem to determine the appropriate model may not be resolved. In order for Zeno’s paradoxes to make the oldest problem of mathematics, you’d need a definite mathematical model of the situation proposed by the ancient Greeks, such that the mathematics of the paradoxes in this particular model is unsolved so far. I don’t see anything like that happening. – Emil Jeřábek May 11 '14 at 14:23
  • 1
    @Emil, I agree with what you wrote with the proviso that the passage from the problem to the model is not as mechanical as your answer tends to suggest. Often it is not clear what the right model is. Part of solving a math problem is determining what the best model is. To take an elementary example, one might think that the Extreme Value theorem is a well-defined topic whose truth value had been resolved long ago. Not so according to constructivists; see this recent article for a discussion. – Mikhail Katz May 11 '14 at 14:27
  • 3
    The sentence of mine that you quote is only asserting that real analysis requires a continuous mathematical axiom, such as the Dedekind completeness axiom, in order to achieve a satisfactory theory. However, this axiom is certainly part of standard mathematical foundations, and I do not believe that any revision of these foundations is necessary in order to perform real analysis. – Terry Tao May 11 '14 at 15:43
  • 4
    My point in the section containing the quote is that from a modern perspective, Zeno's paradoxes can be interpreted as a precursor to the modern theory of the continuum, by highlighting the need for a continuous axiom for the real numbers, as well as the need to specify higher order initial conditions in order for higher order equations of motion to be well posed. – Terry Tao May 11 '14 at 15:48
  • 1
    @Terry, thanks for your comment. I was merely illustrating the mathematical significance of Zeno's paradoxes, a view you seem to share. This is not the place to argue for changing the said foundations, and I am not sure my professional qualifications as differential geometer allow me to argue in favor of changing foundations. The point is that some editors seem to believe that the said foundations provide ultimate answers, which is after all a hypothesis. – Mikhail Katz May 11 '14 at 15:49
  • 6
    I agree that Zeno's paradoxes have mathematical significance. I do not agree that they currently pose a mathematical open problem, which is the focus of the question under discussion. – Terry Tao May 11 '14 at 15:51
  • 1
    @Terry, OK, but nonetheless it cannot be dismissed as being of significance solely as far as physics is concerned. The fact that logicians of the caliber of Keisler are led to provide alternative accounts of it using a hyperreal continuum (based on the same foundations of ZFC) indicates that the paradoxes are richer than is often admitted. – Mikhail Katz May 11 '14 at 15:53
  • 2
    Your original answer in full read: "I was surprised not to find Zeno on this list. Giving a satisfactory mathematical account for resolving Zeno's paradox must surely be the oldest of all." What even is in your opinion the precise unresolved mathematical problem here? If you cannot pin this down, then surely somebody wondered how the starts move and so on before Zeno and surely there is some math there surely some of which can be considered as unresolved. –  May 11 '14 at 18:25
  • 3
    (Edited after 3 hours) I cleaned up by deleting some comments which in my opinion began to veer off-topic and cause tempers to rise, while trying to retain at least most of the core mathematical statements. A few hours have passed and I expect temperatures have lowered, but let's please stay strictly on-topic and strive to be courteous if you wish to say anything more -- thanks. – Todd Trimble May 11 '14 at 19:14
  • 1
    @Todd We would absolutely not mind if you discard our comment (with this one, our two comments) in this thread. – Joël May 11 '14 at 20:14
0

What about the time which elapsed between the question of squaring the circle ( which I was always taught was posed by the Greeks), and the proof that $\pi$ is transcendental in 1882(?) by Lindemann- admittedly not now an open problem, but an impressive time lapse.

Geoff Robinson
  • 42,683
  • Peter Neumann's solution of Alhazen's problem might even outdo this for longevity: http://www-history.mcs.st-and.ac.uk/Obits2/Al-Haytham_Telegraph.html . He found a solution in 1997 to a problem that was posed by Ptolemy around AD150. – Nick Gill Oct 25 '17 at 11:19