# What is the oldest open problem in mathematics?

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?

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Existence of odd perfect numbers? –  Andrey Rekalo Jun 4 '10 at 18:54
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 4 '10 at 18:55
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. code.google.com/p/flyspeck –  Tony Huynh Jun 4 '10 at 19:20
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 4 '10 at 19:23
@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 4 '10 at 19:42

Existence or nonexistence of odd perfect numbers.

Update: Goes back at least to Nicomachus of Gerasa around 100 AD, according to http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html Nichomachus also asked about infinitude of perfect numbers.

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

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Surely you would say that Fermat's Last Theorem (before it was solved) was an open problem dating back to Fermat! –  Noah Snyder Jun 4 '10 at 20:03
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 4 '10 at 21:04
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 4 '10 at 22:17
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 4 '10 at 22:53
@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 5 '10 at 3:42

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 http://www.jstor.org/pss/2320381.

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Not as old or well-defined, but the systematic construction of quasicrystals appeared to be considered around 1200 CE: peterlu.org/content/… –  Steve Huntsman Jun 4 '10 at 19:45

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.

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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 7 '10 at 1:27

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.

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Is there evidence that the Greeks asked this question? –  Gerald Edgar Jun 4 '10 at 22:44
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 4 '10 at 22:53
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 4 '10 at 23:00
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 5 '10 at 1:33
@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 5 '10 at 8:05

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 http://en.wikipedia.org/wiki/Archimedes%27_cattle_problem.

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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 1 '11 at 13:09

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.

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"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 2 '10 at 17:53
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 2 '10 at 18:25
Did EB confuse twin primes with perfect numbers? –  Włodzimierz Holsztyński Apr 11 '14 at 7:43

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 oribits in general triangular (or poligonal) billiards (in the case of irrational angles) remains open. ( http://iml.univ-mrs.fr/editions/preprint2007/files/troubetzkoy_fagnano.pdf ).

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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 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 7:16

The Perfect Cuboid problem was being considered in the early 18th century (according to http://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.

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This is not a big-list question… –  Dirk May 9 '14 at 16:05
'I would guess that in those times they were preoccupied mostly with 2-D problems.' See: en.wikipedia.org/wiki/Doubling_the_cube –  HJRW May 10 '14 at 18:53

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.

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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 http://www.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 http://www.ams.org/mathscinet-getitem?mr=2238333

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OK, I'll bite. Surely this is resolved by the notion of convergence of infinite series? –  HJRW May 8 '14 at 15:39
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 9 '14 at 8:35
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 9 '14 at 9:57
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
(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