Questions tagged [open-problems]

If it turns out that a problem is equivalent to a known open problem, then the open-problem tag is added. After that, the question essentially becomes, "What is known about this problem? What are some possible ways to approach this problem? What are some ways that people have tried to attack it before, and with what results?"

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7 votes
2 answers
2k views

Is there any progress toward solving Gilbreath's conjecture?

Gilbreath's conjecture is a hypothesis, or a conjecture, in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the ...
9 votes
1 answer
2k views

What keeps asymptotic Goldbach's conjecture out of reach of current technology?

Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...
7 votes
1 answer
2k views

Current status of Bloch constant and Landau constant bounds

The Bloch constant B (based on a theorem introduced by André Bloch in 1925 on the maximum radius of a one-to-one disk in the image of a normalized analytic function of the unit disk, see for instance ...
23 votes
3 answers
2k views

Integer-distance sets

Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$. Say that $S$ is an integer-distance set if every pair of points in $S$ is separated by an integer Euclidean distance. ...
31 votes
2 answers
6k views

Is there a known Turing machine which halts if and only if the Collatz conjecture has a counterexample?

Some of the simplest and most interesting unproved conjectures in mathematics are Goldbach's conjecture, the Riemann hypothesis, and the Collatz conjecture. Goldbach's conjecture asserts that every ...
28 votes
3 answers
2k views

Is the ratio Perimeter/Area for a finite union of unit squares at most 4?

Update: As I have just learned, this is called Keleti's perimeter area conjecture. Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and ...
4 votes
1 answer
2k views

If $N = qn^2$ is an odd perfect number with $\gcd(q,n)=1$, is it possible to have $q + 1 = \sigma(n)$?

The title says it all. Question If $N = qn^2$ is an odd perfect number with Euler prime $q$ and $\gcd(q,n)=1$, is it possible to have $q + 1 = \sigma(n)$? Heuristic From the Descartes spoof, with ...
5 votes
1 answer
198 views

On the number of complete Boolean algebras

In their 1972 paper On the number of complete Boolean algebras Monk and Solovay showed that if $\lambda$ is an infinite cardinal, then there are $2^{2^\lambda}$ many isomorphism types of complete ...
8 votes
0 answers
1k views

Is the Collatz conjecture known to be true for interesting unbounded classes of numbers?

The Collatz or the $3n+1$ conjecture is open. Is there a specific polynomial $f(x)\in\mathbb Z[x]$ whose range is unbounded for which every integer of form $|f(m)|$ at $m\in\mathbb Z$ satisfies $3n+1$...
27 votes
8 answers
5k views

Convex hull in CAT(0)

Let $X$ be complete $\mathop{CAT}(0)$-space and $K\subset X$ be a compact subset. Is it true that convex hull of $K$ is compact? Comments: Convex hull of $K$ = intersection of all closed convex sets ...
2 votes
0 answers
196 views

Squares whose differences are squares

EDIT. I've just noticed a thread from 2011 in the "Related" column on the right (click me), where a closely related question is being discussed (the main difference seems to be that, in ...
22 votes
5 answers
4k views

Collection of conjectures and open problems in graph theory

Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?
51 votes
5 answers
5k views

Can $N^2$ have only digits 0 and 1, other than $N=10^k$?

Pablo Solis asked this at a recent 20 questions seminar at Berkeley. Is there a positive integer $N$, not of the form $10^k$, such that the digits of $N^2$ are all 0's and 1's? It seems very unlikely,...
22 votes
4 answers
2k views

Open problems which might benefit from computational experiments

Question: I wonder what are the open problems , where computational experiments might me helpful? (Setting some bounds, excluding some cases, shaping some expectations ). Grant program: The context of ...
2 votes
1 answer
161 views

A more complete set of open problems

Over time, there have been a number of posts on open problems remaining in different fields of math, both here and on the MathSE. So I had the idea of trying to construct a “list of lists” of problems ...
1 vote
0 answers
138 views

Optimal covering trails for every $k$-dimensional cubic lattice $\mathbb{N}^k := \{(x_1, x_2, \dots, x_n) : x_i \in \mathbb{N} \wedge n \geq 3\}$

After a dozen years spent investigating this particular class of problems, I finally give up and I wish to ask you if any improvement is achievable from here on. The general problem is as follows: Let ...
6 votes
2 answers
1k views

5n+1 sequence starting at 7

Consider the following variant of the Collatz function: $f:\mathbb N\rightarrow\mathbb N$ is defined by \begin{equation} f(n):=\begin{cases} n/2 & \text{if $n$ is even}\\ 5n+1 & \...
1 vote
1 answer
289 views

Goldbach conjecture reformulation [closed]

As thought, the question below is a reformulation of the goldbach conjecture. $ S = \{K - ap \mid a \geq 3, p \text{ is prime} < K/2 \} $, where $ a $ is an odd integer greater than or equal to 3, ...
109 votes
29 answers
62k views

Open problems with monetary rewards

Since the old days, many mathematicians have been attaching monetary rewards to problems they admit are difficult. Their reasons could be to draw other mathematicians' attention, to express their ...
7 votes
0 answers
149 views

What is the current status of research on the von Neumann's inequality for $n \ge 3$?

Problem Let $(T_1, \ldots, T_n)$ be a tuple of commuting contractions in Hilbert space $H$. Does a constant $C_n \ge 1$ exist, for which it would be true, that: $$\forall_{p \in \mathbb{C}[x_1, \ldots,...
382 votes
115 answers
105k views

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

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 ...
22 votes
3 answers
978 views

Equilaterally triangulated surfaces with prescribed boundary

There is a problem in Richard Kenyon's list (Wayback Machine) which I would like to post here, because although I have thought about it from time to time, I have not been able to make the slightest ...
0 votes
0 answers
164 views

Research directions related to the Hilbert-Smith conjecture

The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "for every prime $p$ there are no faithful continuous action of the $p$-adic group of integers $A_p$ ...
69 votes
24 answers
18k views

PhD dissertations that solve an established open problem

I search for a big list of open problems which have been solved in a PhD thesis by the Author of the thesis (or with collaboration of her/his supervisor). In my question I search for every possible ...
43 votes
11 answers
15k views

Open questions in Riemannian geometry

What are some major open problems in Riemannian Geometry? I tried googling it, but couldn't find any resources.
106 votes
2 answers
32k views

What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?

In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the ...
66 votes
9 answers
26k views

What are some important but still unsolved problems in mathematical logic?

In the past, first-order logic and its completeness and whether arithmetic is complete was a major unsolved issues in logic . All of these problems were solved by Godel. Later on, independence of ...
2 votes
0 answers
86 views

An open problem about simple Noetherian rings

The following is a well-known open problem in ring theory (see, for instance, Goodearl, Warfield, 'An introduction to noncommutative Noetherian rings, Appendix, Problem 19) Question: Let $R$ be a left ...
4 votes
2 answers
704 views

If all real conjugacy classes are strongly real, then all real irreps are "strongly real"(symmetric), true?

Question Is true that if all real conjugacy classes of a finite group are strongly real, then all its real irreducible representations (irreps) are "strongly real" (symmetric)? And vice ...
37 votes
4 answers
8k views

How to tackle the smooth Poincaré conjecture

The last remaining problem in this whole "everything is a sphere" business, is the smooth Poincaré conjecture in dimension 4: If $X\simeq_\text{homo.eq.} S^4$ then $X\approx_\text{diffeo} S^...
2 votes
2 answers
148 views

On the number of values with exactly $k$ prime factors of a given polynomial

This is surely be a well studied problem. Let $f(x) \in \mathbb{Z}[x]$. Is there some $k \in \mathbb{N}$ such that there are infinitely many $n \in \mathbb{Z}$ where $f(n)$ has exactly $k$ prime ...
4 votes
2 answers
2k views

What is the importance of Polignac’s conjecture?

The twin-prime conjecture (also known as Polignac’s conjecture, 1846) states that there are infinitely many twin primes (pairs of primes that differ by 2; for example, 3 and 5, 5 and 7, 11 and 13, and ...
44 votes
5 answers
23k views

Does pi contain 1000 consecutive zeroes (in base 10)?

The motivation for this question comes from the novel Contact by Carl Sagan. Actually, I haven't read the book myself. However, I heard that one of the characters (possibly one of those aliens at ...
0 votes
0 answers
26 views

The hardness of active learning with fixed budget

I have been looking for theoretical papers studying this question of the fundamental hardness of PAC active learning algorithms. I found a few papers studying the problem from a fixed perspective (...
2 votes
0 answers
283 views

A question on Giles Gardam counter example to the Unit conjecture of Kaplansky

The unit version of the Kaplansky conjecture is about units in $FG$ where $F$ is a field and $G$ is a torsion free group. In a recent counter example by Giles Gardam, it is given an ...
8 votes
0 answers
332 views

Has there been any progress on this open problem about co-well-poweredness of accessible categories?

On the relations between accessible categories and large cardinal axioms, one big example is the following: Assume the existence of a proper class of strongly compact cardinals. Then every accessible ...
15 votes
1 answer
3k views

3-piece dissection of square to equilateral triangle?

At a workshop it was suggested that it likely remains an open problem whether or not there is a 3- or 2 -piece dissection of a square to an equilateral triangle. Can anyone confirm that this is ...
25 votes
7 answers
3k views

Question on consecutive integers with similar prime factorizations

Suppose that $n=\prod_{i=1}^{k} p_i^{e_i}$ and $m=\prod_{i=1}^{l} q_i^{f_i}$ are prime factorizations of two positive integers $n$ and $m$, with the primes permuted so that $e_1 \le e_2 \cdots \le e_k$...
37 votes
0 answers
1k views

Converse of the Archimedean property of the sphere

In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...
17 votes
2 answers
2k views

Why Donaldson's Four-Six Conjecture?

Simon Donaldson apparently made the following conjecture: Two closed symplectic 4-manifolds $(X_1,\omega_1)$ and $(X_2,\omega_2)$ are diffeomorphic if and only if $(X_1\times S^2,\omega_1\oplus\omega)$...
213 votes
0 answers
16k views

Why do polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?

Conjecture. Let $P(x),Q(x) \in \mathbb{R}[x]$ be two monic polynomials with non-negative coefficients. If $R(x)=P(x)Q(x)$ is $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ ...
41 votes
2 answers
5k views

Projective Plane of Order 12

I asked this question on the new Theoretical Computer Science "overflow" site, and commenters suggested I ask it here. That question is here, and it contains additional links, which I doubt I can ...
4 votes
0 answers
172 views

Status of Problems in 102 problems in mathematical logic

Is there any location that records the current status of the problems in 102 problems in mathematical logic? Or, better yet, serves as a status board for open problems in mathematical logic? ...
11 votes
1 answer
1k views

Open problem: $\log n$ factor in Binomial empirical process

The following problem was considered in Cohen and Kontorovich, "Local Glivenko-Cantelli", https://arxiv.org/abs/2209.04054, to appear in COLT 2023 (henceforth, CK'23). Let $Y_j$, $j\in\...
36 votes
5 answers
6k views

Factorials in Pascal's triangle

I asked this question of Keith Conrad, and he suggested that I try posting here. One of my students observed that the only instances of factorials in the interior of Pascal's triangle are $\binom{4}{...
2 votes
0 answers
111 views

An open problem of Hardy and Littlewood on $p$-integral means

In Duren's book "Theory of $H^p$ spaces" (MSN) in the comment section after Section 4, it is mentioned that Littlewood and Hardy proved in Some properties of conjugate functions that if $u$ ...
22 votes
4 answers
3k views

Brute force open problems in graph theory

Usually, a graph theoretic problem asks whether some class of graphs $C$ possesses a quality $P$. For example, $C$ is the class of all graphs and $P$ is the reconstructability property in Kelly-Ulam ...
39 votes
5 answers
3k views

Does there exist a comprehensive compilation of Erdos's open problems?

Fan Chung and Ron Graham's book Erdos on Graphs: His Legacy of Unsolved Problems (A. K. Peters, 1998) collects together all of Erdos's open problems in graph theory that they could find into a single ...
11 votes
1 answer
739 views

Books/blogs/websites that have open problems in Algebraic geometry

I got admitted in a PhD program in Europe last year. But due to serious mental health issues , I was deemed unfit by the mathematics department to continue the program. I am from a 3rd world nation ...
2 votes
0 answers
428 views

What are some of the big open problems in $4$-manifold theory?

I've recently been studying some Manifold Theory and got very interested in their topological as well as geometric properties. From my understanding of the current literature, most the big and ...

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