Not especially famous, long-open problems which anyone can understand

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 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.

Answer 1

One problem which I think is mentioned in Guy's book is the integer block problem: does there exist a cuboid (aka "brick") where the width, height, breadth, length of diagonals on each face, and the length of the main diagonal are all integers?

update 2012-07-12 Since the question has returned to the front page, I'm taking the liberty to add some links that I found after Scott Carnahan's comments. (Scott deserves the credit, really, but I thought the links belonged in the answer rather than in the comments.)

• On perfect cuboids, by Ronald van Luijk, master thesis, 2000.

• The surface parametrizing cuboids, by Michael Stoll and Damiano Testa, arXiv.org:1009.0388.

Answer 2

The moving sofa problem: What rigid two-dimensional shape has the largest area $A$ that can be maneuvered through an L-shaped planar region with legs of unit width?

So far the best results are $2.219531669\lt A\lt 2.37$.

Answer 3

Can we cover a unit square with $\dfrac1k \times \dfrac1{k+1}$ rectangles, where $k \in \mathbb{N}$?

(Note that the areas sum to $1$ since $\displaystyle \sum_{k \in \mathbb{N}}\dfrac1{k(k+1)} = 1$)

Here is an MO thread discussing some of the progress on this problem.

Answer 4

This is the second time I've seen this question on MathOverflow and this will be the second time I've posted this answer.

Singmaster's conjecture says there is a finite upper bound on the number of times a number (other than the $1$s on the edge) can appear in Pascal's triangle. The upper bound may be as low as $8$. If so, then no number (besides those $1$s) appears more than eight times in Pascal's triangle. Only one number is known to appear that many times: $$ \binom{3003}{1} = \binom{78}{2} = \binom{15}{5} = \binom{14}{6} $$

It has been proved that infinitely many numbers appear twice; similarly three times, four times, and six times. It is unknown whether any number appears five times or seven times.

Singmaster states that Erdős said the conjecture is probably true but probably difficult to prove.

Answer 5

The lonely runner conjecture. As Wikipedia puts it:

Consider $k + 1$ runners on a circular track of unit length. At $t = 0$, all runners are at the same position and start to run; the runners' speeds are pairwise distinct. A runner is said to be lonely if at distance of at least $1/(k + 1)$ from each other runner. The lonely runner conjecture states that every runner gets lonely at some time.

Answer 6

The Casas-Alvero conjecture: let the characteristic of the field $k$ be $0$. If a monic polynomial $f\in k[X]$ of degree $n$ has a common root with each of its derivatives $f',\ldots,f^{(n-1)}$, then $f(X)=(X-a)^n$ for some $a\in k$.

Answer 7

Gourevitch's conjecture¹: $$\sum_{n=0}^\infty \frac{1+14n+76n^2+168n^3}{2^{20n}}\binom{2n}{n}^7 = \frac{32}{\pi^3}.$$

¹Jesús Guillera: About a New Kind of Ramanujan-Type Series; Experimental Mathematics (2003), Volume: 12, Issue: 4, page 507-510; DOI: 10.1080/10586458.2003.10504518, eudml

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UPDATE: As mentioned in a comment below, this conjecture has now been proved by K. C. Au, Wilf-Zeilberger seeds and non-trivial hypergeometric identities, arXiv:2312.14051, 26 Dec 2023. But Au mentions another identity that was still conjectural at the time of writing; if we define $(x)_n := x(x+1)(x+2)\cdots(x+n-1)$ then $$\sum_{n=0}^\infty {1920n^2 + 304n + 15 \over 7^{4n}n!^5}\biggl(\frac{1}{2}\biggr)_n \biggl(\frac{1}{8}\biggr)_n \biggl(\frac{3}{8}\biggr)_n \biggl(\frac{5}{8}\biggr)_n \biggl(\frac{7}{8}\biggr)_n = {56\sqrt{7}\over \pi^2}.$$

Answer 8

There is a lot of number theory elementary conjectures, but one that is especially elementary is the so called Giuga Conjecture (or Agoh-Giuga Conjecture), from the 1950: a positive integer $p>1$ is prime if and only if $$\sum_{k=1}^{p-1} k^{p-1} \equiv -1 \pmod{p}$$

Answer 9

Is $e+\pi $ rational?

Answer 10

Is the sequence $(3/2)^n \mod 1$ dense in the unit interval?

In the other direction, Mahler's 3/2 problem:

Do all elements of this sequence with large enough index $n$ lie in the interval $(0,1/2)$?

It is known that $\beta^n$ is uniformly distributed modulo one for almost all $\beta>1$, but explicit examples of $\beta$ for which density holds are not known. This question seems to originate in work of Weyl and Koksma on uniform distribution.

Update: Since posting this answer I've attempted to find some references with which to flesh it out, with only modest success. The earlier paper I have identified which deals with this question directly is T. Vijayaraghavan's 1940 article On the fractional parts of the powers of a number, in which it is shown that the sequence $(3/2)^n \mod 1$ has infinitely many limit points. Mahler conjectured in 1968 that the answer to his question is negative. Jeffrey Lagarias' 1985 survey on the Collatz problem, The 3x + 1 Problem and Its Generalizations, includes a one-page overview of the literature on the distribution of this sequence. Flatto, Lagarias and Pollington subsequently proved that the diameter of the set of accumulation points is at least 1/3; Mahler's question would be answered in the negative if this is improved to "at least 1/2".

Answer 11

It is currently unknown if all triangles have a periodic billiard path. (See, for example, https://en.wikipedia.org/wiki/Outer_billiards#Existence_of_periodic_orbits)

Answer 12

From "An Invitation to Mathematics":

Are there any integer solutions to $x^3 + y^3 + z^3 = 33$?

I thought this might be a good candidate since that book was meant as a bridge from competitive Mathematics to research. There are a few other examples, but I am quoting only one here due to your requirement. Edit: Such integers x, y and z have been found.

Answer 13

I always enjoyed telling people about the Inscribed square problem :

Does every (Jordan) curve in the plane contain all four vertices of some square?

Update: Here is a variation due to Helge Tverberg: Does every (polygonal) curve in the plane outside of the unit circle, contain all four vertices of some square with side length >0.1? This version implies the original problem and lacks disadvantages pointed out by Tim Chow and Henry Cohn. See Ville H. Pettersson, Helge A. Tverberg, and Patric R.J. Östergård, "A Note on Toeplitz' Conjecture," Discrete Comput. Geom. 51 (2014), 722–738.

Answer 14

There are infinitely many primes $p$ such that the repeating part of the decimal expansion of $1/p$ has length $p-1$.

First explicitly asked by Gauss, now generally thought of as a corollary of Artin's primitive root conjecture.

Answer 15

Problem: The partition function $p(n)$ is even (resp. odd) half of the time.

Of course you need to explain to a general audience what the partition function is, but that's not hard, my daughter in K1 got as an assignment to compute $p(n)$ for $n$ up to 4. You also need to explain "half of the time", which means that the number of $n < x$ such that $p(n)$ is even, divided by $x$, has limit 1/2 when $x$ goes to infinity, so you need the notion of limit of a sequence, which is in K12, isn't it ?

The problem is certainly famous among specialists, but not too famous. I don't think there are books on it, for instance. It is old (formulated as a conjecture during the 50th), with an history going back to Ramanajunan. And I like it very much.

UPDATE (28/2/2015) Here is a useful reference:
Ken Ono, The parity of the partition function, Electronic Res. Ann. (1995)

Answer 16

At the risk of stretching my own rule, please allow that I could define "ring" for a high school senior. Then I'd proffer this question I heard years ago from Melvin Henriksen:

Must a non-commutative ring (with identity) contain a non-zero-divisor aside from the identity?

Answer 17

The circulant Hadamard matrix conjecture, first stated in print by Ryser in 1963. It can be stated as follows. If $n>4$, then there does not exist a sequence $(a_1,a_2,\dots,a_n)$ of $\pm 1$'s satisfying $$ \sum_{i=1}^n a_i a_{i+k}=0,\ 1\leq k\leq n-1, $$ where the subscript $i+k$ is taken modulo $n$.

Answer 18

Sendov's Conjecture

For a polynomial $$f(z) = (z-r_{1}) \cdot (z-r_{2}) \cdots (z-r_{n}) \quad \text{for} \ \ \ \ n \geq 2$$ with all roots $r_{1}, ..., r_{n}$ inside the closed unit disk $|z| \leq 1$, each of the $n$ roots is at a distance no more than $1$ from at least one critical point of $f$.

Answer 19

The irrationality of Catalan's constant $G=1-1/3^2+1/5^2-1/7^2+\cdots$.

Remarks: Although Catalan's constant is certainly well-known, the irrationality is the tip of the iceberg of a related conjecture of Milnor about the linear independence over the rationals of volumes of certain hyperbolic 3-manifolds (which is a special case of a conjecture of Ramakrishnan). The irrationality of Catalan's constant would imply that the volume of the unique hyperbolic structure on the Whitehead link complement is irrational. To this date, it is not known that any hyperbolic 3-manifold has irrational volume.

Answer 20

Does the series $\sum_{n=1}^{\infty} \frac{1}{n^3 \sin^2 n}$ converge?

(Taken from https://math.stackexchange.com/questions/20555/are-there-any-series-whose-convergence-is-unknown where there are more such examples)

Answer 21

Here is one which I found at this MO link:

$$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left| \frac{\tan(t)+\sqrt{7}}{\tan(t)-\sqrt{7}}\right|\ dt = \sum_{n\geq 1} \left(\frac n7\right)\frac{1}{n^2}, $$ where $\displaystyle\left(\frac n7\right)$ denotes the Legendre symbol. Not really my favorite identity, but it has the interesting feature that it is a conjecture! It is a rare example of a conjectured explicit identity between real numbers that can be checked to arbitrary accuracy. This identity has been verified to over 20,000 decimal places. See J. M. Borwein and D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, Natick, MA, 2004 (pages 90-91).

P.S. This problem was resolved before this post was placed in Section 5 of [D.H. Bailey, J.M. Borwein, D. Broadhurst and W. Zudilin, Experimental mathematics and mathematical physics, in "Gems in Experimental Mathematics", T. Amdeberhan, L.A. Medina, and V.H. Moll (eds.), Contemp. Math. 517 (2010), Amer. Math. Soc., 41–58]. In fact, the problem was solved even before its mentioning in the 2004 book; the details of the story can be found in the article.

Answer 22

Here are a few others:

1. Let $H_n=\sum_{j=1}^n 1/j$. Then for all $n\geq 1$, $$ \sum_{d|n}d\leq H_n+(\log H_n)e^{H_n}. $$ Jeff Lagarias showed that this is equivalent to the Riemann hypothesis!

2. Let $x_0=2$, $x_{n+1}=x_n-\frac{1}{x_n}$ for $n\geq 0$. Then $x_n$ is unbounded.

3. The largest integer that cannot be written in the form $xy+xz+yz$, where $x,y,z$ are positive integers, is 462. It is known that there exists at most one such integer $n>462$, which must be greater than $2\cdot 10^{11}$. See J. Borwein and K.-K. S. Choi, On the representations of $xy+yz+xz$, Experiment. Math. 9 (2000), 153-158; https://projecteuclid.org/journals/experimental-mathematics/volume-9/issue-1/On-the-representations-of-xyyzzx/em/1046889597.full.

Answer 23

The Kneser–Poulsen conjecture in dimension 3: An arrangement of (possibly overlapping) unit balls in space is tighter than a second arrangement of the same balls if, for all $i$ and $j$, the distance between the centers of ball $i$ and ball $j$ in the first arrangement is less than or equal to the distance between the centers of ball $i$ and ball $j$ in the second arrangement. The conjecture is that a tighter arrangement always has equal or smaller total volume. True in the plane, open in higher dimensions.

Answer 24

Here is another easy to state problem which is 140 years old but not very famous. Consider the potential of finitely many positive charges: $$u(x)=\sum_{j=1}^n\frac{a_j}{|x-x_j|},\quad x,x_j\in R^3,\quad a_j>0$$ How many equilibrium points can this potential have? Equilibrium points are solutions of $\nabla u(x)=0$.

First conjecture: it is always finite.

Second conjecture: when finite, it is at most $(n-1)^2$. This estimate is stated by Maxwell in his Treatease on Electricity and Magnetism, vol. I, section 113, as something known. The editor (J. J. Thomson) wrote a footnote that he "could not find any place where this result is proved".

Nobody could find this place to this time. This is even unknown in the simplest case when all $a_j=1$ and $n=3$.

Answer 25

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 26

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

Answer 27

Does there exist a point in the unit square whose distance to each of the four corners is rational?

This is sometimes called the rational distance problem, although that name often refers to a more general class of similar problems. It's discussed by Richard Guy in Unsolved Problems in Number Theory and in the following paper:

Guy, Richard K. "Tiling the square with rational triangles." Number theory and applications 265 (1989): 45-101.

It's also open whether there's a point outside the square whose distance to each of the four corners is rational, although it is known that no point on the edge of the square has this property.

Answer 28

Schinzel-Sierpinski Conjecture

Taken from this MathOverflow link.

Melvyn Nathanson, in his book Elementary Methods in Number Theory (Chapter 8: Prime Numbers) states the following:

• A conjecture of Schinzel and Sierpinski asserts that every positive rational number $x$ can be represented as a quotient of shifted primes, that $x=\frac{p+1}{q+1}$ for primes $p$ and $q$. It is known that the set of shifted primes, generates a subgroup of the multiplicative group of rational numbers of index at most $3$.

Answer 29

A few decades ago Sherman Stein asked whether a trapezoid whose parallel sides are in the ratio $1:\sqrt2$ can be dissected into triangles, all of the same area. This remains open--it's a mystery which trapezoids admit such dissections.

Answer 30

Erdos's problem on the length of lemniscates (it is somewhat famous in certain narrow circles). Let $P$ be a polynomial, and consider the set $E=\{ z:|P(z)|=1\}$ in the complex plane.

What is the maximum length of $E$ over all monic polynomials of degree $d$?

Erdos conjectured that an extremal $P$ is $P_0(z)=z^d+1$.

It is known that the asymptotic of maximal length is $2d+o(d).$ It is known that $P_0$ gives a local maximum. It is also known that for every extremal polynomial, all critical points lie on $E$, so $E$ must be connected.

However the conjecture is not established even for $d=3$.

After Erdos's death, I offered a $200 prize for the first solution. (Erdos had offered the same, but I do not know whether one can collect his prize.)