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Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.

Motivation: I plan to use this list in my teaching, to motivate general education undergraduates, and early year majors, suggesting to them an idea of what research mathematicians do.

Meaning of "not too famous" Examples of problems that are too famous might be the Goldbach conjecture, the $3x+1$-problem, the twin-prime conjecture, or the chromatic number of the unit-distance graph on ${\Bbb R}^2$. Roughly, if there exists a whole monograph already dedicated to the problem (or narrow circle of problems), no need to mention it again here. I'm looking for problems that, with high probability, a mathematician working outside the particular area has never encountered.

Meaning of: anyone can understand The statement (in some appropriate, but reasonably terse formulation) shouldn't involve concepts beyond high school (American K-12) mathematics. For example, if it weren't already too famous, I would say that the conjecture that "finite projective planes have prime power order" does have barely acceptable articulations.

Meaning of: long open The problem should occur in the literature or have a solid history as folklore. So I do not mean to call here for the invention of new problems or to collect everybody's laundry list of private-research-impeding unproved elementary technical lemmas. There should already exist at least of small community of mathematicians who will care if one of these problems gets solved.

I hope I have reduced subjectivity to a minimum, but I can't eliminate all fuzziness -- so if in doubt please don't hesitate to post!

To get started, here's a problem that I only learned of recently and that I've actually enjoyed describing to general education students.

https://en.wikipedia.org/wiki/Union-closed_sets_conjecture

Edit: I'm primarily interested in conjectures - yes-no questions, rather than classification problems, quests for algorithms, etc.

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    $\begingroup$ Here is some collection of some other "collect open problems" quests. on MO: mathoverflow.net/questions/96202/… PS Nice question ! PSPS may be add tag "open-problems" $\endgroup$
    – Alexander Chervov
    Commented Jun 21, 2012 at 20:53
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    $\begingroup$ Nice question!! $\endgroup$
    – Suvrit
    Commented Jun 22, 2012 at 3:25
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    $\begingroup$ To save the search for explanation of cryptic acronyms for those of us outside US, K-12 means high school. @Mahmud: You are using a wrong meaning of the word “problem”. The TSP is not an unproved mathematical statement, it is a computational task. $\endgroup$
    – Emil Jeřábek
    Commented Jun 22, 2012 at 12:05
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    $\begingroup$ More precisely, K-12 means anything up to high school (K = Kindergarten, 12 = 12th grade, and K-12 covers this range). $\endgroup$
    – Henry Cohn
    Commented Jun 22, 2012 at 13:05
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    $\begingroup$ There seems to be a claimed proof of the union-closed sets conjecture by Blinovsky arxiv.org/abs/1507.01270 $\endgroup$
    – Marco
    Commented Oct 22, 2015 at 14:08

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What is the least $S$ (if any) such that any subset of a plane of area $S$ contains $3$ vertices of a triangle of unit area?

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I really like this problem of Busemann and Petty (Problems on convex bodies, Math. Scand., 1956):

Given a $0$-symmetric convex body $K$ and one of its support hyperplanes, construct the solid cone whose apex is a point at which the hyperplane touches the body and whose base is the central section of $K$ that is parallel to the hyperplane. Clearly the volume of this cone will be independent of the choice of contact point (in the case the support hyperplane intersects the body in more than one point). Now assume that no matter what support hyperplane you choose, the volume of the cone is always the same. Is $K$ an ellipsoid?

As far as I know there are not even meaningful partial solutions to this problem (e.g., assume $K$ is three-dimensional and the boundary of $K$ is real analytic and strictly convex).

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In number theory, the Odd greedy expansion problem concerns a method for forming Egyptian fractions in which all denominators are odd.

Stein, Selfridge, Graham, and others have posed the question of

whether the odd greedy algorithm terminates with a finite expansion for every $x/y$ with $y$ odd?

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I like the Montesinos-Nakanishi 3-move conjecture from knot theory. A 3-move on a link is the replacement of two parallel strands by three half twists. The conjecture is that any link can be turned into the trivial link by a sequence of such moves. (If you think of this as a conjecture on diagrams, then you also need to allow Reidemeister moves.) According to an Encyclopedia of Mathematics article:

The conjecture has been proved for links up to 12 crossings, 4-bridge links and five-braid links except one family represented by the square of the centre of the 5-braid group. This link, which can be reduced by 3-moves to a 20-crossings link, is the smallest known link for which the conjecture is open (as of 2001).

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The Kurepa problem: Show that for all primes $p>3$ we have that $$ 0!+1!+2!+\dots+(p-1)! $$ is not divisible by $p$. Kurepa posed this problem in 1971. For an overview see the article by Ivic and Mijajlovic (https://arxiv.org/abs/math/0312202).

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    $\begingroup$ Already on this list: see mathoverflow.net/a/114639/763 $\endgroup$
    – Yemon Choi
    Commented Dec 17, 2014 at 16:18
  • $\begingroup$ Is there a particular reason to believe that this problem has a positive answer? -- Naively, for large enough $n$ I would expect about $\ln(\ln(n))$ counterexamples less than $n$. But maybe there is a particular reason why this heuristics is not applicable here? $\endgroup$
    – Stefan Kohl ♦
    Commented Dec 17, 2014 at 16:30
  • $\begingroup$ @StefanKohl you can try search for n=exp(exp(3)) if you find this large enough. $\endgroup$
    – joro
    Commented Dec 17, 2014 at 16:53
  • $\begingroup$ @Yemon Choi: Sorry, I didn't see that there is a second page. $\endgroup$
    – Jan-Christoph Schlage-Puchta
    Commented Dec 18, 2014 at 14:05
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    $\begingroup$ @Stefan Kohl: Ivic told me that the Barsky-Benzaghou-proof is philosophically correct in the sense that although it does not prove the conjecture, it gives a reason why the conjecture should be true. I haven't looked at the paper itself, though. $\endgroup$
    – Jan-Christoph Schlage-Puchta
    Commented Dec 18, 2014 at 14:08
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  • Is Hilbert's tenth problem for Diophantine equations in rational numbers decidable?
  • Is Hilbert's tenth problem for Diophantine equations of power $3$ decidable?
  • Is there a universal Diophantine equation of power $3$?
  • Is there a universal Diophantine equation containing less than $9$ variables? If so, what is the minimal number of variables? What minimal power can be achieved for that number of variables?
  • Is there a universal Diophantine equation that can be written using less than $100$ arithmetic operations (additions or multiplications)? If so, what is the minimal number of operations?
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What is the least $V$ such that any convex body of unit volume can be fit into a tetrahedron of volume $V$? It is known that $V \ge 9/2$ and conjectured that $V = 9/2$.

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Is there any odd perfect number?

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    $\begingroup$ Methinks this one is both pretty famous and long open... $\endgroup$
    – José Hdz. Stgo.
    Commented Jun 14, 2013 at 15:21
  • $\begingroup$ Well, maybe too famous, but I was not sure. $\endgroup$
    – The User
    Commented Jun 14, 2013 at 23:05
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    $\begingroup$ answer all mathematicians are both odd and perfect at the same time $\endgroup$
    – Riemann-bitcoin.
    Commented Apr 18, 2017 at 14:19
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A conjecture arising from Waring problem: a number of solutions of the equation $$x_1^3+x_2^3+x_3^3=y_1^3+y_2^3+y_3^3,\qquad|x_i|,|y_i|<P,$$ is $O(P^{3+\varepsilon})$. The best known estimate is $O(P^{7/2+\varepsilon})$ (Hua Loo Keng).

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More than ten years ago I posed the following problem in a couple of math-related mailing lists:

Let $G_n$ be the graph with vertex set $\{1, 2, \dots, 2n\}$ such that $\{i,j\}$ is an edge if and only if $i+j$ is a prime number. Is it true that $G_n$ is Hamiltonian for every $n \geq 2$?

It is a simple consequence of Bertrand's Postulate (there is always a prime between $k$ and $2k$) that $G_n$ is connected and has a perfect matching for every $n$.

The problem turned out to be an old one. I believe that some variation of it appears in Richard K. Guy's "Unsolved Problems in Number Theory" and according to this article, it was originally posed in the Journal of Recreational Mathematics in 1982.

Michael A. Jones and Leslie Cheteyan, "Two observation on unsolved problem #1046 on prime circles of $\{1, 2, . . . , 2m\}$", J. Recreational Mathematics Vol.35(1) (2006), 15--19.

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    $\begingroup$ Surely you don’t meen Eulerian? For example, $G_4$ isn’t Eulerian as it has many odd degree vertices. $\endgroup$
    – OHO
    Commented Mar 3, 2020 at 7:36
  • $\begingroup$ I meant "Hamiltonian". Thanks for pointing that out. $\endgroup$
    – Santi Spadaro
    Commented Mar 23, 2020 at 21:53
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    $\begingroup$ I can't find it in Guy's book. A recent paper is Chen, Fu, and Guo, From a consequence of Bertrand's postulate to Hamiltonian cycles, arxiv.org/abs/1804.07104 The authors prove the conjecture holds for infinitely many $n$. They seem to be unaware of previous discussions of the conjecture. $\endgroup$
    – Gerry Myerson
    Commented Mar 23, 2020 at 23:47
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    $\begingroup$ Also, the question has been discussed here on MO: mathoverflow.net/questions/241569/… where Zhi-Wei Sun gives the reference Antonio Filz [J. Recreational Math. 14(1982), p.64] and Max Alekseyev gives the reference oeis.org/A051252 – oh, and it is in Guy's book, in the section (C1) on Goldbach's conjecture, with the attribution to Filz. $\endgroup$
    – Gerry Myerson
    Commented Mar 23, 2020 at 23:59
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    $\begingroup$ Another reference is Stan Wagon's Problem of the week 1218, mathforum.org/wagon/current_solutions/s1218.html $\endgroup$
    – Gerry Myerson
    Commented Mar 24, 2020 at 0:11
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I like Moser's worm problem: https://en.wikipedia.org/wiki/Moser%27s_worm_problem

I'm not sure how famous it is, but I believe less than it deserves. In short: what is the area of the smallest planar region such that every curve of unit length can be placed into it?

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  • $\begingroup$ Is it even clear that there's a lower bound in the non-convex case? I feel like if I had to bet on a number I'd probably pick 0 by analogy with Kakeya... $\endgroup$
    – Steven Stadnicki
    Commented Aug 18, 2021 at 19:32
  • $\begingroup$ I faintly remember that the problem appeared in the German version of Scientific American where it was called "mother worms" problem $\endgroup$
    – Manfred Weis
    Commented Nov 9, 2022 at 16:13
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If $2^x$ and $3^x$ are integers for some positive real number $x$, does this imply that $x\in\mathbb{N}$?

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    $\begingroup$ See also this answer. $\endgroup$
    – Alex Ravsky
    Commented Oct 10, 2022 at 2:51
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Does every nonseparating planar continuum have the fixed point property?

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    $\begingroup$ For those of us whose general topology is rusty: what is the K-12 level formulation of this question? $\endgroup$
    – Yemon Choi
    Commented Jun 21, 2012 at 22:50
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    $\begingroup$ For those rusty on general topology, if K is a closed and bounded subset of the (x,y) plane, if K has connected and simply connected complement U, and if f:K-->K is continuous, must K have a fixed point? To aim at K-12, PL arcs in U and `uniform continuity' are adequate to reformulate the respective properties of U and f in an elementary manner. $\endgroup$
    – Paul Fabel
    Commented Jun 22, 2012 at 13:05
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    $\begingroup$ Uniform continuity is a K-12 concept now? $\endgroup$
    – Douglas Zare
    Commented Jun 22, 2012 at 14:25
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    $\begingroup$ Thanks Yemon. I agree with Doug that uniform continuity is not a stand alone K-12 concept. I mentioned it in order to avoid trying to decode its meaning in K-12 terms in this context, as follows: $\endgroup$
    – Paul Fabel
    Commented Jun 22, 2012 at 17:56
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    $\begingroup$ f is a set of 4-tuples (x1,y1,x2,y2) so that each point (x1,y,1) in K appears precisely once as the 1st two coordinates of some point in f, and we also require that the last two coordinates of each point of f is a point of K. For each positive radius R we can find a smaller positive radius r(R) so that if (x1,y1,x2,y2) is in f, then if (w1,z1) is both in K and also in the disk of radius r(R) centered at (x1,y1), and if (w1,z1,w2,z2) is in f, then (w2,z2) is in the disk of radius R centered at (x2,y2). $\endgroup$
    – Paul Fabel
    Commented Jun 22, 2012 at 17:56
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(From Alon and Tarsi)

Let $A$ be a nonsingular $n$ by $n$ matrix over the finite field $\bf{F}_q$, $q>4$, then there exists a vector $x$ in $\bf{F}_q^n$ such that both $x$ and $Ax$ have no zero component.

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    $\begingroup$ It was proved to be true if q is not a prime or if q is prime and n is bounded by a function of q. $\endgroup$
    – Joe Franklin
    Commented Dec 8, 2017 at 12:04
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Can a square with integer sides contain a point whose distance from each corner is an integer?

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    $\begingroup$ D19 in Guy, Unsolved Problems in Number Theory, with many references. $\endgroup$
    – Gerry Myerson
    Commented Mar 24, 2020 at 0:14
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    $\begingroup$ Is this Belk's problem above, June 2015? $\endgroup$
    – Ben McKay
    Commented Jun 24, 2020 at 10:20
  • $\begingroup$ Yes, it is. (I just noticed Belk's contribution.) $\endgroup$
    – Mickybo Yakari
    Commented Jun 24, 2020 at 17:54
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Are there infinitely many partition numbers divisible by $3$? See A000041.

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  • $\begingroup$ Could you perhaps expand a bit, i.e. for example what makes this a difficult problem, and what is known about the same question for divisors other than $3$? $\endgroup$
    – Stefan Kohl ♦
    Commented Nov 9, 2015 at 10:12
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    $\begingroup$ For example, it is known that there are infinitely many divisible by 2. It is hard for me to estimate how difficult the problem is, but it was publicly raised at OEIS more than 10 years ago (and I'm sure it had been discussed before), it is still open, and it is about a famous sequence that received much attention (and new interesting results) during last decade. The sequence graph suggests that the distribution of such numbers is quite irregular, they do not seem to to lie nicely on a smooth curve. $\endgroup$
    – Vladimir Reshetnikov
    Commented Nov 9, 2015 at 17:23
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Does the decimal expansion of $2^n$ contain the digit $7$ for all sufficiently large $n$?

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    $\begingroup$ Is there anything specific about base 10 or digit 7? $\endgroup$
    – R. van Dobben de Bruyn
    Commented Aug 19, 2021 at 16:14
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N. M. Katz: "Simple Things we don't know": https://web.math.princeton.edu/~nmk/pisa16.pdf

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  • $\begingroup$ These will be a better fit here: mathoverflow.net/questions/101169/… when the collective net gods deem it time to reopen the question. $\endgroup$
    – David Feldman
    Commented Jul 14, 2012 at 19:07
  • $\begingroup$ These will be a better fit here: mathoverflow.net/questions/101169/ when the collective net gods deem it time to reopen the question. Every vote helps, thanks. $\endgroup$
    – David Feldman
    Commented Jul 14, 2012 at 19:08
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I get the feeling that you will enjoy reading about the Simonyi and Chvatal conjectures described here by some guy called Gil Kalai. Anyone know who that is? ;)

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The continuum hypothesis. Of course it's extremely famous, but everyone thinks it's resolved. I was astonished to find out that some serious set theorists apparently consider it (I mean in the present, decades past Cohen's proof) to be an important open problem that people should be working on solving (for some meaning of "solve").

P. Koellner ( http://logic.harvard.edu/EFI_CH.pdf, Wayback Machine ) describes some current approaches.

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    $\begingroup$ You seem to allude that it has not been resolved yet. But in that case you should explain what you mean by that. On the other hand, the continuum hypothesis is so well-known that it does not fit to the question. $\endgroup$
    – Martin Brandenburg
    Commented Jul 19, 2012 at 12:17
  • $\begingroup$ +1 because the note you linked is really interesting! Although the "problem" is rather open-ended, I think it could be stated in a more well-defined way. Here's a stab at it: "If we want to settle the continuum hypothesis, which axioms should we add to ZFC in order to do it?" $\endgroup$
    – Vectornaut
    Commented Jul 22, 2012 at 18:24
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    $\begingroup$ By the way, I think your statement that "everyone thinks it's resolved" is a little misleading. Maybe it would be better to say it this way: "most people think there's nothing left to say about it, because it's been proven independent of ZFC." $\endgroup$
    – Vectornaut
    Commented Jul 22, 2012 at 18:26
  • $\begingroup$ this is not an open problem. It has been settled (like many impossibility results) a long time ago. $\endgroup$
    – andrey bovykin
    Commented Sep 10, 2019 at 4:08
  • $\begingroup$ I have added a Wayback Machine link. It seems that the same file can be found in some other places - of course, it's hard to say whether some of those links will remain stable. $\endgroup$
    – Martin Sleziak
    Commented Nov 9, 2022 at 10:55
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"Which regular polygons are constructible with a marked straightedge/ruler and compasses?"

This was posed by Baragar in 2002 after demonstrating that some (but not necessarily all) quintic equations can be solved with the above tools [1]. It follows that constructions for regular $n$-gons exist not only if the Euler totient of $n$ has a maximum prime factor of $2$ or $3$ such as $n=5$ or $n=13$ (with $2$ as the maximum prime factor, construction is also possible without marks on the straightedge), but maybe if the Euler totient has a maximum factor of $5$ like $n=11$ or $n=25$. The problem has since been solved (the answer is "yes") for $n=11$ (and thus also for $11$ times numbers with Euler-totient prime factors $\le3$), but not for $25,31$, or any other "quintic" cases.

Reference

  1. Baragar, A. (2002). "Constructions Using a Compass and Twice-Notched Straightedge". The American Mathematical Monthly, 109(2), 151–164. https://doi.org/10.2307/2695327
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Here are 2 problems related to cyclotomic polynomials that are unsolved :

Let $n,m>1$ and $$A(x) = \Phi_n(x) + \Phi_m(x)$$

  1. If $n,m$ are distinct odd primes then $A(x)$ is irreducible.

  2. If $A(x)$ factors then one of its factors is a cyclotomic polynomial.

Example :

$$\Phi_7(x) + \Phi_{22}(x) = \Phi_4(x) b(x)$$

$$\Phi_4(x) = x^2 +1 $$ $$ b(x) = x^8 - x^7 + 2 x^4 + 2$$

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  • $\begingroup$ This is interesting, but do you have a reference to somewhere these conjectures are discussed? $\endgroup$
    – Sam Hopkins
    Commented Jan 2 at 23:09
  • $\begingroup$ @SamHopkins Apparently it has been checked up to $m,n < 151$ in the year $2000$ by some "Nicol ". I am unaware of theoretical ideas or numerical tests after the year $2000$. I have not seen good arguments for or against. $\endgroup$
    – mick
    Commented Jan 2 at 23:20
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    $\begingroup$ When it comes to cyclotomic polynomials, that's not a very big range. See mathoverflow.net/a/15506/25028. $\endgroup$
    – Sam Hopkins
    Commented Jan 2 at 23:21
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    $\begingroup$ Without more references I fear this one fails the requirement that "[t]he problem should occur in the literature or have a solid history as folklore." $\endgroup$
    – Sam Hopkins
    Commented Jan 2 at 23:43
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    $\begingroup$ @GerryMyerson no. statement 1 and 2 are distinct ! ( i had this debate before but the poster removed the talk ) $\endgroup$
    – mick
    Commented Dec 11 at 14:31
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Ore's odd Harmonic number conjecture.

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    $\begingroup$ I suspect this fails on the fame criterion. Gerhard "Ask Me About System Design" Paseman, 2012.06.27 $\endgroup$
    – Gerhard Paseman
    Commented Jun 27, 2012 at 14:22
  • $\begingroup$ Yes, but Ore's odd harmonic number conjecture, which if true, implies no odd perfect numbers, seems just right here. $\endgroup$
    – David Feldman
    Commented Jun 28, 2012 at 4:35
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    $\begingroup$ Well, now it's so obscure that it requires a context/explanation. $\endgroup$
    – Victor Protsak
    Commented Jan 6, 2014 at 20:45
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Denote by $\sigma(m)$ be the sum of divisors of $m$. When is $ \sigma(n!-1) $ a perfect square?

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    $\begingroup$ Is this a long-open problem? Does it have a history? $\endgroup$
    – Gerry Myerson
    Commented Sep 15, 2017 at 23:02
  • $\begingroup$ Numbers k such that σ(k) is a square are tabulated at oeis.org/A006532 – it is not even known that there are infinitely many of them (although this would follow from standard conjectures) $\endgroup$
    – zeraoulia rafik
    Commented Oct 17, 2017 at 19:17
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    $\begingroup$ Isn't jstor.org/stable/10.4169/… proving that there are infinitely many? $\endgroup$
    – Dima Pasechnik
    Commented Jan 12, 2019 at 14:23
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$P(x)=x^2+1$. $\mathbb P$ the set of prime number.

  1. Is it true that $P(\mathbb N)\cap \mathbb P$ is an infinite set ?

  2. Can you find a $Q \in \mathbb Z[x]$, with $\deg(Q)\geq 2$ and $Q(\mathbb N) \cap \mathbb P$ is an infinite set ?

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    $\begingroup$ This seems like a famous open problem since it is the 4th of Landau´s four problemshttps://en.wikipedia.org/wiki/Landau%27s_problems . $\endgroup$
    – JoshuaZ
    Commented Mar 5, 2023 at 13:44
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