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John Stillwell, in his textbook on arithmetic cites Erdos:

As the great Hungarian problem-solver Paul Erdos liked to point out, if you can think of an open problem that is more than 200 years old, then it is probably a problem in number theory.

What are some examples of >200 years old unsolved problems NOT from number theory?

Rodrigo de Azevedo
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Alexandre Eremenko
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    The Riemann Hypothesis, which we may think of as a problem in complex analysis, is a youngster: only 163 years old. – Gerald Edgar Aug 27 '22 at 00:47
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    Irrationality of Euler's constant $\gamma$ and odd zeta values $\zeta(n)$ with $n>3$, see Lagarias. The definitions go back to Euler (1730s). I do not think he explicitly asked about their irrationality, perhaps because he had no approach to proving that. But he did prove irrationality of $e$ in 1737, and wrote that he had no way of "expressing" odd zeta values in a letter. – Conifold Aug 27 '22 at 20:24
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    Fagnano noticed in 1775 that in every acute triangle there is a periodic billiard trajectory, it is still open whether the same holds for obtuse triangles, see Halbeisen-Hungerbuhler, On Periodic Billiard Trajectories in Obtuse Triangles. – Conifold Aug 27 '22 at 20:53
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    @Conifold: I asked for problems NOT in number theory. Some number theory problems go back to more than 2000 years ago, and they are well known. Same applies to the Riemann hypothesis (this is not a problem of complex analysis). – Alexandre Eremenko Aug 28 '22 at 12:04

2 Answers2

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(This probably does not answer your question, so it can be moved to the comments.)

The fourth book of Albrecht Dürer's Vnderweysung der messung (1525), is devoted (among other things) to polyhedra, and in particular he explores the possibility to give a planar "unfolding" of a polyhedron (here the truncated cuboctahedron). Dürer's list is not exhaustive, and so one can argue that he (implicitely) posed the problem to give a planar unfolding (with no overlaps) for a general polyhedron. This lead to Dürer's problem, also known as Shephard's conjecture (1975), which is an open problem.

Now, our problem is the following: does Dürer's problem date back to 1525, or is it simply a modern reinterpretation inspired by his work?

EDIT

Some comments and a partial answer:

I believe that there are fundamentally two difficulties in finding old open problems in mathematics, and in particular not related to number theory.

The first is that many problems, initially of a geometric or combinatorial nature, can be reformulated in numerical terms and in fact become number theory problems: I am thinking, for example, of the problem of finding a perfect cuboid, or the three great problems of antiquity, including the problem of finding the regular polygons that can be constructed with a ruler and compass (which is equivalent to determining all of Fermat's primes).

The second is that, even in the case of number theory problems, it is not easy to establish when and by whom a certain problem was actually posed. For instance, Nicomachus of Gerasa is often attributed with certain (still open) problems concerning perfect numbers. In reality it is doubtful that his contemporaries thought those were problems: rather, they were considered as facts. Only in the 16th century Hudalrichus Regius showed that not all numbers of the form $2^n-1$ for all primes $n$ are themselves prime, and so being able to give a counterexample to one of Nicomachus' statement.

This situation is similar to the one cited with regard to Dürer: although he had all the tools he needed to formulate a conjecture (and thus to pose a problem), he probably simply thought it was natural that it was always possible to unfold a polyhedron, and indeed in the fourth book cited he provides a 'practical' algorithm (here I'm citing the second Latin edition of Dürer's treatise by Camerarius):

Si quis igitur ea componere velit, is accipiat duo folia papyri, bitumine, aut pasta cohaerentiaa, et corpora illa super eadem ita describat, ut acuto cultello alterum foliorum secundum lineas ductas scindatur, et cum omnia reliqua ex residuo papyri fuerint soluta, tunc facile complicabitur in ductibus et scissuris. Ideoque advertendum est ad sequentem ductinem, nam talia ad plurima conducunt.

i.e.

So if someone wants to make them up, take two sheets of paper, some bitumen or a cohesive paste, and describe those bodies on them in such a way that with a sharp knife one of the sheets is torn according to the lines drawn, and when all the rest of the paper has been unbound, then it will be easily folded according to the lines and cuts. For this, it is necessary to pay attention to the following directions, because these lead to different solutions.

The same can also be said of Fagnano's problem. In fact, the problem formulated and solved by Fagnano is actually the following: "In a given acutangle triangle, inscribe a triangle whose perimeter is the smallest possible." For example, in Two Applications of Calculus to Triangular Billiards by Eugene Gutkin, (The American Mathematical Monthly, Vol. 104, No. 7, 1997), we read:

We assume that the triangle T is acute, and view it as a billiard table. The pedal triangle T_1, which is inscribed in T, is then a periodic billiard orbit [...] Moreover [...] is the shortest [...], the only closed (prime) billiard orbit known [...] The first proof, by calculus, that among all inscribed traingles the pedal triangle has the least perimeter, is attributed to J.F.F. Fagnano, ca. 1775. In his honor, the problem just stated is often called the Fagnano problem [...]

It is therefore clear that in reality Fagnano did not pose the problem, not only in its full generality (i.e., in the case of any polygon), but not even in the case of a non-acutangle triangle (which is still an open problem, one of the 5 most resistant problems in dynamics, according to Anatole Katok).

Returning to the original question, one answer (cheating a bit) could be the Kepler conjecture: today it is no longer considered an open problem, but when (2003) the first edition of Stillwell's book was published, I think it still was (in 1998, Thomas Hales announced that he had a proof, but only in 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof, accepted by the journal Forum of Mathematics, Pi in 2017).

In the same spirit (an open problem at the time of the first edition of Stillwell's book), the Ptolemy's problem (or Alhazen's problem, and somehow related to the Fagnano's problem), i.e. "find the point on a spherical convex mirror at which a ray of light coming from a given point must strike in order to be reflected to another point," does not fit in for only 6 years: Alhazen gave the first geometric solution in his Book of Optics, and algebraic solutions were found in the 20th century, but only in 1997 was proved that there is no ruler-and-compass construction to solve it.

user6530
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  • I suppose that that 1997 from the final paragraph is a reference to the article by Peter Neumann, but this was only a rediscovery. The first person who showed that the problem has no solution was Jack M. Elkin (“A deceptively easy problem”, Mathematics Teacher, 58 (3): 194–199), in 1965. – José Carlos Santos Aug 29 '22 at 07:36
  • The link to "5 most resistant problems in dynamics" does not work. – Alexandre Eremenko May 28 '23 at 06:56
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As a possible example "of >200 years old unsolved problems NOT from number theory" there may be the following:

According to Peter Pesic, "[V I] Arnol’d points out [that] Leibniz was moved to conjecture that (in modern terms) an Abelian integral along an algebraic curve (taken between algebraic limits) is a transcendental number. This conjecture remains an important unsolved problem." (Historia Mathematica 28 (2001), 215–219, at 218).

I didn't succeed in tracing the source for this in Arnol'd or in Leibniz, {edit:} but @user6530 has since then kindly supplied a link to the original form of Leibniz's conjecture, given in a letter of 1691. The subject of the 1691 discussion and conjecture appears to be integrability, and there is no mention of transcendental numbers (a concept which also appears to have been in 1691 still in the future). From this it appears that the so-called "modern terms" of the conjecture put forward in the secondary source (which was all that I originally had available to quote), may be a more radical alteration of the original than only a rendering of it into 'modern terms', and it may be unclear whether it fairly represents the original meaning and scope of Leibniz's 1691 conjecture. Indeed my lack of any primary source was the reason why I mentioned the conjecture only as a "possible" item within the category requested in the question. In the current state of the discussion, it does not seem clear that Leibniz's calculus problem falls within number theory. But a clarification of the claimed scope of 'number theory' for the purposes of this question does seem desirable. Whether the conjecture offered here 'falls within' number theory might depend on whether number theory is now being defined so wide as to include any problem in any field for which some other problem can be specified which is in some way parallel with it and also arguably falls within number theory. Such a definition might seem to colonize the whole of mathematics.

terry-s
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    The original letter of Leibniz (in French), is here: https://www.dbnl.org/tekst/huyg003oeuv10_01/huyg003oeuv10_01_0022.php – user6530 Sep 02 '22 at 12:03
  • And here https://arxiv.org/abs/1805.10104 the authors claim to have answered the Leibniz's problem – user6530 Sep 02 '22 at 12:07
  • Why a conjecture about trancendence of some numbers is "not from number theory"? – Alexandre Eremenko Sep 02 '22 at 13:46
  • @Alexandre Eremenko : Because the conjecture appears primarily to arise in the field of calculus with perhaps some geometry -- albeit that it can be and has been restated or translated with reference to number. – terry-s Sep 02 '22 at 13:55
  • @terry-s: Problems of number theory arise everywhere, for example, Pythagorean triples, or constructions with compass and ruler arise in geometry. But they are number theory problems. – Alexandre Eremenko Sep 02 '22 at 15:02
  • @Alexandre Eremenko : I grant you the pythagorean triples (assuming you mean those composed of integers). As for problems in geometry or calculus (e.g. that do not refer to number, as for example in the original form of Leibniz's conjecture kindly supplied by user6530), what criterion do you assert for subsuming them into number theory? Did Erdos also assert such a criterion? Would you consider clarifying the question accordingly? – terry-s Sep 02 '22 at 16:41
  • Problems about proving that certain numbers are transcendental certainly belong to number theory. – Alexandre Eremenko Sep 02 '22 at 17:07
  • @AlexandreEremenko Not according to standard terminology. "Number" in number theory traditionally refers to integers only, and irrationality/transcendentality problems are typically listed under analysis, as their formulation and solution usually involves analysis. You should specify your expanded notion of "number theory" in the OP to avoid unwanted entries. – Conifold Sep 02 '22 at 22:28
  • @Conifold: this is a very strange opinion to me that "transcendental numbers" is a chapter of Analysis. In the standard mathematical classification (MSC codes), transcendental numbers (11J, 11K) are listed under "number theory" (11xxx). – Alexandre Eremenko Sep 03 '22 at 14:25
  • @Alexandre Eremenko : My original answer did say that the explanation of the mentioned conjecture was in a secondary source and "in modern terms". The source kindly supplied by user6530 shows that the original form, dating from 1691, was very different and does not mention transcendental numbers. Such a concept seems in 1691 still to have been in the future. So it does not seem possible to decide the question whether or not Leibniz's conjecture belongs to number theory purely on the basis of a reference to 'transcendental number' which does not appear to be part of the original conjecture. – terry-s Sep 04 '22 at 00:54