Question

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 (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.

http://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.

Answer 1

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 eulerian 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.

The whole issue can be downloaded here:

http://www.baywood.com/comppdf/0022-412x.pdf

Answer 2

Let ${^n a}$ denote tetration: ${^0 a}=1, {^{n+1} a}=a^{({^n a})}$.

• It is unknown if ${^5 e}$ is an integer.
• It is unknown if there is a non-integer rational $q$ and a positive integer $n$ such that ${^n q}$ is an integer.
• It is unknown if the positive root of the equation ${^4 x}=2$ is rational (ditto for all equations of the form ${^n x}=2$ with integer $n>3$)
• It is unknown if the positive root of the equation ${^3 x}=2$ is algebraic.

Answer 3

Waring's problem inequality

One of the oldest (Since 1770) and famous open problem in number theory is Waring's problem. It has been conjectured that if

$$ Frac\bigg[\bigg(\frac{3}{2}\bigg)^n\bigg] \le 1 - \bigg[\bigg(\frac{3}{4}\bigg)^n\bigg]. $$

(where $Frac$ denotes the fractional part) true then, the general solution of Waring's problem is

$$ g(n) = 2^n + Int\bigg[\bigg(\frac{3}{2}\bigg)^n\bigg] - 2. $$

Answer 4

The Kurepa conjecture : For every odd prime $p$, one has $$ 0!+1!+\cdots+(p-1)!\not\equiv0\pmod p $$ A proof was claimed and published in 2004 but the claim was withdrawn in 2011. See also my comment on the accepted answer to MO24265.

Answer 5

The Polya--Szego conjecture for polygonal drums: among the polygonal drums with $n$ sides and given area, the regular one has the slowest vibration (and therefore the lowest tone).

As far as I know, this remains open for $n\geq 5$.

Answer 6

Let $R(x)=P(x)/Q(x)$ where $P(x)$ and $Q(x)$ are polynomials with integer coefficients and $Q(0)\neq 0$. Is there an algorithm that given $P(x)$ and $Q(x)$ as an input always halts and decides if the Taylor series of $R(x)$ at $x=0$ has a coefficient $0$?

Answer 7

Bonnesesn—Fenchel conjecture: Which body of constant width has the least volume? Is it Meissner's tetrahedron?

Answer 8

Is there a rectangle that can be cut into $3$ congruent connected non-rectangular parts?

Answer 9

Is there a positive integer which is both triangular and factorial except these obvious examples: $1, 6, 120$? (Tomaszewski conjecture, http://oeis.org/A000217)

Answer 10

• Is the Ring of Periods actually a field? (most likely, no)
• Is the equality of periods decidable? (hopefully, yes)

Answer 11

What is the largest possible volume of the convex hull of a space curve having unit length?

Answer 12

This is basically copied from my answer on this question, which I've now updated some.

Let's let $\|n\|$ denote the smallest number of 1's needed to write n using an arbitrary combination of addition and multiplication. For instance, ||11||=8, because $11=(1+1)(1+1+1+1+1)+1$, and there's no shorter way. This is sequence A005245.

Then we can ask: For n>0, is $\|2^n\|=2n$?

Since it is known that for m>0, $\|3^m\|=3m$, we can ask more generally: For n, m not both zero, is $\|2^n 3^m\|=2n+3m$?

Attempting to throw in powers of 5 will not work; ||5||=5, but $\|5^6\|=29<30$. (Possibly it could hold that $\|a^n\|=n\|a\|$ for some yet higher choices of a, but I don't see any reason why those should be any easier.)

Jānis Iraids has checked by computer that this is true for $2^n 3^m\le 10^{12}$ (in particular, for $2^n$ with n≤39), and Joshua Zelinsky and I have shown that so long as $n\le 21$, it is true for all m. (Fixed powers of 2 and arbitrary powers of 3 are much easier than arbitrary powers of 2!) In fact, using an algorithmic version of the method in the linked preprint, I have computed that so long as $n\le 41$, it is true for all $m$, though I'm afraid it will be some time before I get to writing that up...

That seems to be the best known.

Answer 13

The following conjecture by Carsten Thomassen:

If $G$ is a 3-connected graph, every longest cycle in $G$ has a chord.

Thomassen has proven the conjecture true for 3-connected cubic graphs.

Answer 14

Enumeration of meanders. (See also meander).

Problem is to find some formula for the number of meanders or at least some good asymptotic.

As far as I understand the attention to it has been attracted by V.I. Arnold. The problem is so "everyone can understand" that there is an article by him in the math. journal for shool-children "Quant" (sorry it is in Russian. I remember it from my school years): djvu file from the site.

There are plenty papers in arXiv on the problem.

E.g. http://arxiv.org/abs/cond-mat/0003008 Exact Meander Asymptotics: a Numerical Check Philippe Di Francesco, Emmanuel Guitter (SPHT-Saclay), Jesper Lykke Jacobsen (LPTMS-Orsay)

As far as I understand from the nice book (or) by S. Lando and A. Zvonkin the problem is still open.

Answer 15

• 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?

Answer 16

Is there such $n\in\mathbb{N}$ that ${^n\pi}\in\mathbb{N}$? (see tetration)

Answer 17

Is there an upper bound of quotients in the continued fraction representation of $\sqrt[3]{2}=[ 1; 3, 1, 5, 1, 1, \dots]$?

Answer 18

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?

Answer 19

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 $V = 9/2$.

Answer 20

Is there a dense subset of a plane having only rational distances between its points?

Answer 21

Is there a triangle that can be cut into $7$ congruent triangles? (no)

Answer 22

From Rick Kenyon's open problem list:

What are the minimal number of squares needed to tile an $a \times b$ rectangle?

Kenyon showed the correct order is $\log a$ assuming $a/b$ is bounded with $b \leq a$. However, there is plenty of room for improvement in the constant factor, and an exact formula seems far, far away.

Answer 23

How many trees are there?

Let $T(n)$ be the number of trees on $n$ vertices up to graph isomorphism. There is no known closed formula for $T(n)$.

In 1947 Richard Otter proved^[Source] the asymptotic result $$T(n) \sim A \cdot B^n \cdot n^{-\frac{5}{2}}$$ where $A \approx 0.535$ & $B \approx 2.996$.

By way of contrast, let $L(n)$ be the number of labelled trees, i.e. trees formed from vertices labelled $1,...,n$ where isomorphism additionally preseves the label. In 1889, Arthur Cayley showed^[Source] that $$L(n)=n^{n-2}$$

Answer 24

The following problem is very well-known among algebraic geometers:

Does there exist a cubic 4-fold that is not rational?

It's probably not well-known outside of algebraic geometry, even though it can easily be explained in every elementary terms:

Does there exist a polynomial equation $F$ of degree three in five variables with the following property: Let $X \subset \mathbb C^5$ be the solution set of $F = 0$. Then there exists no chart $U \subset \mathbb C^4, \phi \colon U \to X$ such that $\phi$ is defined by rational functions (i.e., quotients of polynomials).

Answer 25

The complexity of matrix multiplication (i.e. the asymptotic number of steps required to multiply two n-by-n matrices).

This is an important problem in CS theory, but is non-famous enough in other fields that a mathematician (Andrew Stothers) made a significant advance in it in 2010 (beating a 20-year-old bound of Coppersmith and Winograd), and wrote up the result on page 71 of his PhD thesis without bothering to state it as a theorem or otherwise call attention to it. Word of it only got around a year or so later, when a computer scientist (Virginia Vassilevska Williams) independently made a further improvement.

The obvious multiplication algorithm takes $O(n^3)$ steps, and a well-known Karatsuba-like rearrangement gets the exponent $\omega$ down to about 2.8. There is a simple proof that the smallest possible $\omega$ is $\ge 2$. Coppersmith and Winograd got an exponent of 2.376 and the more recent results have it at 2.373. Apparently nobody has even shown that the minimum is not equal to 2: there are some who believe there's an algorithm faster than $O(n^{2+\epsilon})$ for any $\epsilon>0$ but not an $O(n^2)$ algorithm, but this is not known.

More info is in this blog post of Scott Aaronson: http://www.scottaaronson.com/blog/?p=839

Answer 26

http://en.wikipedia.org/wiki/Keller%27s_conjecture

From Wikipedia:

Keller's conjecture is the conjecture introduced by Ott-Heinrich Keller (1930) that in any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face.

Keller's original cube-tiling conjecture remains open in dimension 7.

Conjecture was shown to be true in dimensions at most 6 by Perron (1940a, 1940b). However, for higher dimensions it is false, as was shown in dimensions at least 10 by Lagarias and Shor (1992) and in dimensions at least 8 by Mackey (2002), using a reformulation of the problem in terms of the clique number of certain graphs now known as Keller graphs. Although this graph-theoretic version of the conjecture is now resolved for all dimensions, Keller's original cube-tiling conjecture remains open in dimension 7.

The related Minkowski lattice cube-tiling conjecture states that, whenever a tiling of space by identical cubes has the additional property that the cube centers form a lattice, some cubes must meet face to face. It was proved by György Hajós in 1942.

Szabó (1993), Shor (2004), and Zong (2005) give surveys of work on Keller's conjecture and related problems.

Answer 27

Do there exist five positive integers such that the product of any two of them increased by 1 is a perfect square?

The same question for seven distinct nonzero rationals.

Diophantine m-tuples pages

Answer 28

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 ) describes some current approaches.

Answer 29

Alexander's Conjecture, and by extension a lot of open problems about combinatorial subdivision, are as easy to understand as they are maddening. To quote Melikhov:

Alexander's 80-year old problem of whether any two triangulations of a [3-dimensional] polyhedron have a common iterated-stellar subdivision. They are known to be related by a sequence of stellar subdivisions and inverse operations (Alexander), and to have a common subdivision (Whitehead). However the notion of an arbitrary subdivision is an affine, and not a purely combinatorial notion. It would be great if one could show at least that for some family of subdivisions definable in purely combinatorial terms (e.g. replacing a simplex by a simplicially collapsible or consructible ball), common subdivisions exist...

Stellar subdivision (and arbitrary subdivisions) can be explained to a K-12 student with a picture. For a stellar subdivision, choose a face F, take its midpoint, and connect it to all vertices of tetrahedra of which F is a face. For arbitrary subdivision, invent some silly triangulation of a simplex, and just plug it inside. refining heighbouring simplexes as needed.

Answer 30

N. M. Katz: "Simple Things we don't know": https://web.math.princeton.edu/~nmk/pisa16.pdf