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The question of "What is the oldest open problem in mathematics?" comes up from time to time, and there seems to be consensus that the answer is "Are there any odd perfect numbers?".

There are many other old open problems in number theory. See the prior mathoverflow question on that question for some examples.

What I'm interested to know here is:

What is the oldest (or some candidates for the oldest) open math problem clearly outside of number theory?

EDIT: The same question was asked two years ago on the History of Science and Mathematics StackExchange, but did not elicit an accepted answer.

This is a soft question, so I do not want to apply too rigid of 'rules'. But a few guidelines are:

  • I can't formally define what is number theory. Certainly I want to exclude anything to do with primes, factorizations, irrationality, transcendentality, constructibility, rational/integral solutions to equations, etc.
  • There should be a clear record of the problem being formulated as a conjecture or question (as opposed to "so-and-so surely would have considered X after studying y".)
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    $\begingroup$ see hsm.stackexchange.com/q/14748/1697 $\endgroup$
    – Carlo Beenakker
    Commented Sep 13 at 6:30
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    $\begingroup$ I’m voting to close this question because it has already been asked on hsm.se, where it is more on-topic $\endgroup$
    – Federico Poloni
    Commented Sep 13 at 9:12
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    $\begingroup$ @FedericoPoloni I am not sure. I expect that there exist many mathematicians who can answer questions like this and who are active rather on MO than on HSM $\endgroup$
    – Fedor Petrov
    Commented Sep 13 at 11:18
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    $\begingroup$ @FedericoPoloni if mathematicians are not against to see such questions, then why not? History of mathematics is intimately related to mathematical research: every research paper has historical part; the high interest to history of mathematics is much more natural for mathematicians then to anybody else, and it can teach us a lot. The same concerns philosophy of mathematics, teaching mathematics and other things mathematicians have a natural interest in. $\endgroup$
    – Fedor Petrov
    Commented Sep 13 at 12:29
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    $\begingroup$ @TimothyChow If it is properly answered on HSM, then it can be closed as a duplicate, but I have doubts about this. Many questions from Math.se are duplicated here if they do not get an answer there, why should it be different with HSM? $\endgroup$
    – Fedor Petrov
    Commented Sep 13 at 14:11

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I don't have a 200 year old candidate but I can offer a 150 year old one (from Maxwell's "Treatise on Electricity and Magnetism" written in 1873).

You are allowed to put $n$ electric point charges in $\mathbb R^3$ in any way you want. What is the maximal number of equilibrium points of the resulting electric field as a function of $n$? (there are several variation of that: both positive and negative charges are allowed, only positive charges are allowed, all charges must be unit, you place them in $\mathbb R^d$ with arbitrary $d$, etc., etc.).

As far as I know, in full generality the question still remains open. I learned it from Alexandre Eremenko, who can probably comment more on it.

This was just to set the "trivial lower bound" :-)

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    $\begingroup$ It seems (from the paper of Gabrielov, Novikov and Shapiro: arxiv.org/pdf/math-ph/0409009) that Maxwell asserted without proof an upper bound (in fact $(n-1)^2$) on the number of equilibrium points in his treatise, rather than posing it as a question/conjecture. That said, it is a good example particularly given that it is precisely formulated in the original source. $\endgroup$
    – Mark Lewko
    Commented Sep 14 at 16:58
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    $\begingroup$ @Mark Lewko: more precisely, they did not prove finiteness, but assuming finiteness, they proved un upper bound. As far as I know, the problem is unsolved even for 3 charges. I've a computer assisted for 3 equal charges, but did not verify that it is correct, and it is not published. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 15 at 13:10
  • $\begingroup$ I still have flashbacks about this problem - Boris Shapiro gave me this problem to work on, first year as an undergraduate student. I did not manage to get far :) $\endgroup$
    – Per Alexandersson
    Commented Oct 3 at 18:10
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Stability of the Solar System ?

(Question often attributed to Newton in Opticks, 1717 or 1730.)

To further specify as requested by Timothy Chow, make it a few ($3\leqslant N\leqslant 8$) planets under pure Newtonian attraction, as in Suzuki, A history of the stability problem in celestial mechanics, from Newton to Laplace (1642–1787) (p. 24):

The other form of the stability question was that the planets themselves, by mutual gravitation alone, might disturb their orbits “until the system wants a reformation,” a point most famously raised by Newton in Optics. Because of the uncertainty of the validity of universal gravitation, this problem could not even be asked until after mid-century, but thereafter, progress was rapid and by 1760, a preliminary answer was obtained by Charles Euler, followed, over the next two decades, by the works of Lagrange and Laplace.

A “clear record of the problem being formulated as a conjecture or question” is in e.g. Laplace (1784):

si l’on n’a égard qu’aux lois de la gravitation universelle, les moyennes distances des corps célestes aux foyers de leurs forces principales sont immuables (…). Mais les excentricités et les inclinaisons sont-elles renfermées constamment dans d’étroites limites? C’est un point important du système du monde qui reste encore à éclaircir.

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    $\begingroup$ I was about to suggest this one, but it's a little murky. First, is it a physics question about the actual solar system, or a math question about the equations of Newtonian mechanics? If the latter, what is the question exactly? E.g., Arnold's work on the KAM theorem tells us that there are some solar-system-like systems that are stable. So maybe the question is determining which systems are stable and which aren't. In addition to the article you linked to, this Quanta article is also interesting. $\endgroup$
    – Timothy Chow
    Commented Sep 14 at 12:36
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    $\begingroup$ That question, or some version of it, is still open, but it isn't the question Newton asked. This article by Scott Tremaine gives some more information about what the early discussions of the problem looked like. $\endgroup$
    – Timothy Chow
    Commented Sep 14 at 12:41
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    $\begingroup$ Suzuki (1996, §1.5 “The Stability Question Raised”) discusses the two versions: “non-gravitational causes”, versus analysis of the purely gravitational problem (p. 24). The latter is the mathematical problem I have in mind, discussed by Laskar. $\endgroup$
    – Francois Ziegler
    Commented Sep 14 at 12:47
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    $\begingroup$ Right, but precisely how many masses are we including in our mathematical idealization of the solar system? Just the eight planets? Or do we include dwarf planets? How close do the masses, positions, and velocities in our mathematical idealization have to be to the masses of the actual physical objects they're modeling? Arnold's result concerns "solar systems" with parameter values that are many orders of magnitude different from those in our actual solar system. At what point are we asking questions about the space of all possible solar systems rather than "the" solar system? $\endgroup$
    – Timothy Chow
    Commented Sep 14 at 13:30
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    $\begingroup$ There is actually a bizarre story surrounding the "Charles Euler" paper. Apparently Euler entered a Paris Academy prize competition in his son's name (Carl/Carolus/Charles). Jacobi speculates that this was because he had been teased for winning these competitions so many times. For proof of the true author Jacobi cites the inscription: "The father decided that all the stars in heaven should move." $\endgroup$
    – N M
    Commented Sep 16 at 0:33
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I suggested this candidate in a comment: For which $d$ and $g$ does there exist a curve in $\mathbb{P}^3$ of degree $d$ and genus $g$? In Hartshorne's Algebraic Geometry, Chapter VI, Section 6, it is indicated that the Steiner Prize was awarded in 1882 to Noether and Halphen for work on this problem and related problems (although if one wants to be pedantic about the earliest date on which the question was formulated explicitly, I don't know when that would be). According to another MO answer, Hartshorne's claim that this problem is open remains true today.

If this particular problem doesn't answer Mark Lewko's question, then there may be similar questions in classical algebraic geometry that do. For comparison, the interpolation problem, of determining when there is a (Brill–Noether) curve of degree $d$ and genus $g$ passing through $n$ general points in $\mathbb{P}^r$, was only recently solved by Larson and Vogt.

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  • $\begingroup$ Look at the reference I gave hsm.stackexchange.com/questions/14748/… for examples "outside physics" and older than those in this list. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 15 at 13:13
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    $\begingroup$ @TimothyChow: do you have any information about the original source of this problem? As an aside, I had never heard of the Steiner Prize. Apparently for some of its existence (which came to an end with WWI) eminent mathematicians of the era (including Dirichlet and Weierstrass) would pose problems, with the idea the prize would go to the best work on the problem. Reportedly these problems frequently went unsolved and the prizes unclaimed. It would be interesting to know if the full statements of these problems survive. See: ams.org/bookstore/pspdf/mprize-prev.pdf $\endgroup$
    – Mark Lewko
    Commented Sep 15 at 22:44
  • $\begingroup$ @MarkLewko I don't think that tracking down the text of the people who offered the prize is going to lead anywhere. A more promising route may be to look at the collected works of Halphen, to see what problems he stated explicitly. $\endgroup$
    – Timothy Chow
    Commented Sep 15 at 22:49
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The Navier–Stokes equation was mentioned in a comment so I thought I would give a link to Sylvio R. Bistafa's essay, 200 Years of the Navier-Stokes Equation, which gives some historical information. Briefly, although Navier's work appeared in 1822, the Clay Prize version of the problem is probably not older than the problem fedja suggested, because the early work on the equation implicitly or explicitly assumed laminar flow.

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    $\begingroup$ I originally voted to close this MO question as a duplicate of the HSM question, but since the community has decided to reopen the MO question, I edited the HSM question to point here to MO. $\endgroup$
    – Timothy Chow
    Commented Sep 14 at 13:11
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The inverse Galois problem is another possible candidate, though it may not be easy to track down the earliest explicit statement of it. Wikipedia claims that it was posed in the early 19th century, but Galois died in 1832 and there does not seem to be any concrete evidence that he explicitly posed the problem. I kind of doubt it was explicitly posed any earlier than 1853, when Kronecker first stated what we now call the Kronecker–Weber theorem. In the other direction, the problem should be dated no later than Hilbert's 1892 Crelle paper, "Uber die Irreducibilitat ganzer rationaler Functionen mit ganzzahligen Coefficienten".

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    $\begingroup$ This problem would almost always be considered part of number theory. $\endgroup$
    – Will Sawin
    Commented Sep 15 at 23:14
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    $\begingroup$ @WillSawin Could be, though in my anecdotal experience, number theorists don't seem to be particularly interested in it, while (for example) group theorists like John Thompson are. $\endgroup$
    – Timothy Chow
    Commented Sep 15 at 23:20
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    $\begingroup$ Well for example David Zywina has done a lot of work on it. $\endgroup$
    – Will Sawin
    Commented Sep 16 at 12:33
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    $\begingroup$ ... and, after all, who gets to decide what is or isn't "number theory"? :) $\endgroup$
    – paul garrett
    Commented Sep 16 at 17:27
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    $\begingroup$ Number theorists are very much interested in the quantitative variant (how many number fields of given $G$ and discriminant at most $X$ in absolute value), cf Malle's conjecture and work on it. $\endgroup$
    – Aurel
    Commented Sep 17 at 18:43
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A long-standing unresolved problem in celestial mechanics is to determine the number / finiteness of relative equilibria in the $n$-body problem, for any choice of positive masses $m_1, \ldots, m_n$. (#6 of Smale's problems.) The $n$-body problem was introduced at least as far back as Newton, but does anyone have a reference for the earliest reference to this specific question?

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