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

Ramanujan's conjecture [*] If $2^x$ and $3^x$ are both rational (hereafter assumed) integers for some non-zero $x$ then $x$ is an integer.

[*] I think that is the accepted name for this problem. He certainly proved the weaker corresponding result with $2^x$, $3^x$, and $5^x$ all assumed to be integers.

Unlike some of the other fascinating conjectures already listed here, this one seems "obviously" true. Yet I gather little progress has been made on it. It must be hard to find a foothold, so to speak, or know where to start.

Another easily understood example is the Erdős-Straus Conjecture which asserts that for every integer $n > 1$, there is at least one set of positive integers $x, y, z$ with $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{4}{n}$. The result is trivially true if negative integers are also allowed.

In this case, by contrast, it's easy(ish) to "almost" prove it, and with patience and ingenuity one can proceed (apparently) ever closer to a solution. But a few annoying special cases always seem to slip through the net!

One more example - I think a high school kid would have little difficulty understanding the abc conjecture, or following the simple proof of the corresponding result for polynomials Mason-Stothers theorem.

Answer 2

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 3

I think nobody pointed this problem, if it is repeated, please say me to delete it. This problem killed me for three weeks, when I was a young student in high school. So, I want to recall it again.

Problem: Find all right triangles with rational sides, where the area of these triangles are integer?

I think it is still open problem and if somebody can solve it, I will give 100$ as a small award.

After I searched, I found these two interesting sources. I hope it will be helpful.

1) N.Koblitz, Introduction to elliptic curves and modular forms, volume 97 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1993.

2) Washington, Lawrence C., Elliptic Curves : Number Theory and Cryptography, CRC Press Series On Discrete Mathematics and Its Applications

Answer 4

The Cerny conjecture says that if X is a collection of mappings on an n element set such that some iterated composition (repetitions allowed) of elements of X is a constant map then there is a composition of at most $(n-1)^2$ mappings from X which is a constant mapping. This comes from automata theory. See https://en.wikipedia.org/wiki/Synchronizing_word.

Answer 5

Proving the Inequality of the Means by fitting boxes into a cube. From Berlekamp, Conway and Guy's Winning Ways for Your Mathematical Plays, Academic Press, New York 1983. See the discussion of this problem on Dror Bar-Natan's webpage for details, pictures, etc.

Question: Is it possible to pack $n^n$ rectangular n-dimensional boxes whose sides are $a_1, a_2,\ldots, a_n$ inside one big n-dimensional cube whose side is $a_1+a_2+\cdots+a_n$?

Answer 6

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 7

The Graph Reconstruction Conjecture:

Let $G, H$ be finite, simple, loopless graphs such that $|V(G)|$ and $|V(H)|$ are at least $4$. If there is a bijection $\varphi:V(G)\to V(H)$ such that for all $v\in V(G)$ the graphs $G\setminus \{v\}$ and $H\setminus \{\varphi(v)\}$ are isomorphic, then $G\cong H$.

Answer 8

Are there eight points on the plane, no three on a line, no four on a circle, with integer pairwise distances?

The analogous question for seven points was posed by Paul Erdős and answered positively by Kreisel, Kurz 2008, who have then asked this question.

In general, problems by Paul Erdős are worth to check if you want to find problems you are asking for here.

Tobias Kreisel, Sascha Kurz, There Are Integral Heptagons, no Three Points on a Line, no Four on a Circle, Discrete & Computational Geometry 39/4 (2008), 786-790. (Wayback Machine)

Answer 9

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: https://scottaaronson.blog/?p=839

Answer 10

An open problem I find surprising, the PAC (Perimeter to Area Conjecture) due to Keleti (1998):

The perimeter to area ratio of the union of finitely many unit squares in the plane does not exceed 4.

See for example Bounded - Yes, but 4? and references therein.

Answer 11

Here's another Birch Swinnerton-Dyer related problem. Sylvester conjectured that every prime that is 4,7 or 8 mod 9 is a sum of two rational cubes. Elkies (unpublished?) settled the first two cases. As far as I know, the third is still open.

Answer 12

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

Answer 13

For integers $a_1, a_2, \ldots$, let $[a_1,a_2,\ldots]$ denote a continued fraction expansion $a_1 + \frac{1}{a_2 + \frac{1}{\ddots}}$.

Zaremba conjectured in 1972 that there must exist some constant $A > 1$ such that given any integer $d > 1$, you can always find some coprime $b$ such that if we write out the continued fraction expansion $b/d = [a_1, a_2, \ldots, a_k]$, all of the coefficients $a_i \leq A$.

Everyone seems to believe that $A = 5$, but we still don't know if such a constant exists. The current best known result (of Bourgain and Kontorovich) is that almost all integers $d$ permit a $b$ such that $b/d$ has the desired property (in their paper, they took $A = 50$, but I think that has been improved slightly since).

Answer 14

The Littlewood conjecture:

For any $\alpha, \beta \in \mathbb{R}$ we have $$\lim\textrm{inf}_{n\to\infty} (n\cdot||n\alpha||\cdot||n\beta||) = 0$$

where $||\cdot||$ denotes the distance to the nearest integer.

Answer 15

Can a disk be dissected into two or more congruent pieces, with its centre lying within one of the pieces?

Answer 16

The following is a conjecture of Wlodzimierz Kuperberg:

Every convex planar set of area 1 is contained in a quadrilateral of area $1+\frac{4}{5}\tan\frac{\pi}{5}\sin\frac{\pi}{5}$.

In other words, such a set is contained in a quadrilateral of area less that $\sqrt{2}$, and the minimum is obtained for the minimum area quadrilateral containing a regular pentagon.

The conjecture involved only elementary plane geometry, and can be found in:

W. Kuperberg, On minimum area quadrilaterals and triangles circumscribed about convex plane regions, Elem. Math. 38 (1983), no. 3, 57–61, MR0703939 (85a:52009)

It is presented as a challenge to the MO community here:

Small quadrilaterals containing a given convex region

---

It is easy to prove that

(*) Every convex planar set of area 1 is contained in a quadrilateral of area 2.

It is also easy to see that statement (*) remains true if the constant 2 is replaced with a somewhat smaller one. Contest: Find such a constant, the smaller the better.

Update:

Reaching $\sqrt{2}$ and even a strictly smaller value was proved by Chakerian (1973) and Kuperberg (1983) and the research challenge offered is to improve it even further, and perhaps even to verify the conjecture that the minimum is attained by a regular pentagon. But any nice arguments for bounds below 2 are welcome.

Answer 17

3D Version Of Blaschke-Lebesgue(1914) Theorem

The planar, compact.convex set of constant width, say 1, of minimal area is the Reuleaux triangle: Blaschk-Lebesgue(1914). The 3D set of constant width and minimal volume is unknown.

Answer 18

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

Answer 19

Is it true that any word of length $n$ contains less than $n$ squares?

(A square is a factor of the form $uu$ for a non-empty word $u$.)

Answer 20

This problem is open (to my knowledge), surely accessible to anybody, and not too famous. As for the "Long open" requirement, there is at least a sizable group of people interested in it, as evidenced by this MO post: Factoring 0-1 polynomials

The problem is:

Is it true that if a polynomial $p(x)$ whose coefficients are in $\{0,1\}$ factors as $p(x)=p_1(x)p_2(x)$, where $p_1$ and $p_2$ are monic polynomials with non-negative real coefficients, then in fact also $p_1$ and $p_2$ have coefficients in $\{0,1\}$?

Answer 21

The Feit-Thompson conjecture is not too famous and it is still open: there are no prime numbers $p\neq q$ such that $$\frac{p^q-1}{p-1}\text{ divides }\frac{q^p-1}{q-1}.$$

Answer 22

Here is a nice question due to John Conway. In a magical 4x4 square, show that the XOR composition of the four numbers, written in base 2, in every row and in every column is zero. This applies to a square in which the numbers 0 to 15 are used (rather than 1 to 16).

For instance, a typical row might be 0 15 14 1, which in binary is 0000 1111 1110 0001, and in each of the four positions there happen to be two entries 0 and two entries 1, so the binary sum is zero.

Of course there are only finitely many possible magic 4x4 squares, and you can give proof by "complete inspection" (aka brute force). In fact, that has been done, so the result is true. But neither he nor I know a conceptual proof. Should be easy to understand about a classical problem -- and yet seems not obvious. Try it!

(Incidentally, the binary sum along the diagonals need not always be zero; that's not part of the question.)

Answer 23

Are there infinitely many $n\ge1$ such that $$ \gcd(2^n-1,3^n-1)=1 ? $$ Ailon and Rudnick conjectured that the answer is affirmative around 2000. What I like about this problem is that it could appear in Euclid, yet wasn't asked until fairly recently. There are obvious generalizations, and the analogue with $\mathbb Z$ replaced by $\mathbb C[T]$, is proven in the Ailon-Rudnick paper. There are also some (deep) results of Bugeaud, Corvaja, and Zannier giving the following related results:

• There is a constant $C>0$ and infinitely many $n\ge1$ such that $$ \gcd(2^n-1,3^n-1) \ge \exp(C n/\log\log n) . $$
• For every $\epsilon>0$ there is a $C_\epsilon$ such that $$ \gcd(2^n-1,3^n-1) \le C_\epsilon\exp(\epsilon n) \quad\text{for all $n\ge1$.} $$

Answer 24

Imre Ruzsa conjectured in 1971 (Mat. Lapok 22, in Hungarian) that a congruence-preserving mapping $f : \mathbb{N} \to \mathbb{Z}$ is a polynomial as soon as the power series $A(t) := \sum_{n \in \mathbb{N}} f(n)t^n \in \mathbb{Z}[[t]]$ has radius of convergence $> 1/e$. (Congruence-preserving simply means $n-m \mid f(n)-f(m)$.)

This is still an open problem, although A. Perelli and U. Zannier have shown that the power series $A(t)$ must be $D$-finite ("On recurrent mod $p$ sequences," J. reine angew. Mat. 348, 1984, DOI: 10.1515/crll.1984.348.135, eudml). The best result on Ruzsa's problem is due to U. Zannier ("On periodic mod $p$ sequences and G-functions," Manuscripta math. 90, 1996, eudml, DOI: 10.1007/BF02568314).

Answer 25

In an oriented graph, is there always a vertex from which there are at least as many vertices that one can access by moving along exactly two edges, than there are vertices that one can access by moving along one edge?

This is known as Seymour's second neighborhood conjecture, and might be on the verge to being too famous (but it seems few of my colleagues know it).

Answer 26

Ron Graham [1,2] asked if there are infinitely many positive integers $n$ such that the central binomial coefficient $\binom{2n}{n}$ is coprime to $105 = 3 \times 5 \times 7$, and offered a prize of $1.000 for a proof/disproof.

Accordingly to some heuristics [3, §4], there should be infinitely many such $n$, but if instead of $105$ a product of four primes is taken, than only finitely many such $n$ are expected.

Furthermore, it has been proved [4] that $\binom{2n}{n}$ if coprime to $pq$ for infinitely many $n$, where $p$ and $q$ are two fixed odd primes.

[1] D. Berend and J. E. Harmse, On some arithmetic properties of middle binomial coefficients, Acta Arith. 84 (1998), 31–41.

[2] OEIS, https://oeis.org/A030979

[3] C. Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, 112 (2015), 636-644.

[4] P. Erdős, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp. 29 (1975), 83-92.

Answer 27

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 28

Here is another problem on equilibrium points of potentials: suppose that we have infinitely many point masses in $R^3$ (the points do not accumulate). Must there exist a point where the gravitational force created by these masses is zero?

If the masses $m_k>0$ are placed at $x_k\to\infty$ then the force is $$\sum_{k=1}^\infty m_k\frac{x-x_k}{|x-x_k|^3},\quad \mbox{where}\quad\sum_{k=1}^\infty m_k|x_k|^{-2}<\infty.$$ Does every such function have a zero?

This is a version of the problem proposed by Lee Rubel in 1980-th. For some partial results see https://www.math.purdue.edu/~eremenko/dvi/equil.pdf This question can be easily modified for any dimension $n\geq2$, using Newtonian ($n\geq 3$) or logarithmic potential ($n=2$). The question is substantially easier in dimension $2$, but even for $n=2$ it is not solved in full generality.

Answer 29

Is the density of $1$s in the Kolakoski sequence $122112122122112112212112\dots$ (Wikipedia, OEIS) equal to $1/2$? Also, does every consecutive block, which occurs at all in the Kolakoski sequence, occur infinitely often?

Answer 30

A list of fundamental geometry problems with simple intuitive statements involving curves and surfaces in Euclidean space is maintained at:

Open Problems in Geometry of Curves and Surfaces

Five of these problems are listed below in no particular order (see the paper for references, background, and more precise statements):

Durer's Unfolding Problem: Can every convex polyhedron be cut along a collection of its edges and developed into the plane in a single non-overlapping piece?

Alexandrov's Conjecture on Intrinsic Diameter: Of all convex surfaces with a fixed intrinsic diameter (the distance between the farthest two points as measured within the surface), the one with the greatest area is the doubled disk (the degenerate convex surface obtained by gluing a pair of disks along their boundaries).

Zalgallers's Problems on Width and Inradius: What are the shorted closed curves in Euclidean 3-space with a given width or inradius? (The width is the smallest distance between all pairs of parallel planes which contain the curve in between them, and the inradius is the radius of the largest ball which is contained in the convex hull of the curve and is disjoint from the curve; see the post by Joseph O'Rourke).

Euler-Maxwell Rigidity Problem: Can one continuously deform a smooth closed surface in Euclidean 3-space without changing the distances between its points, as measured within the surface?

Converse of the Archimedes Hatbox Theorem: Is the sphere the only convex surface such that whenever it is cut by a pair of parallel planes separated by a fixed distance, the area of the portion of the surface trapped between the planes is always the same? See this earlier post.