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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34 votes
9 answers
3k views

Places where one can post open problems

(This must have been asked before and exist somewhere in Community Wiki, but I can't find it...) Where can you post open (math) problems? And what are the advantages and disadvantages? Example: This ...
7 votes
0 answers
394 views

Landau's century-old problems: Anything comparable?

Landau's four problems are now over a century old (1912), and each still unsolved. This seems remarkable, even though he was not the originating author all four (maybe only the 4th?). Still, he ...
0 votes
1 answer
94 views

Traveling salesperson problem algorithm [closed]

I was wondering something, let's say in a symmetric distance matrix of a sample of TSP, there was a sure algorithm that could remove between 30% to 80% of the values (distances) that wouldn't ...
1 vote
1 answer
153 views

Infinitely many $k \in \mathbb{N}$ such that the closed interval $[k, k+99]$ contains from $2$ to $23$ prime numbers

Let $k \in \mathbb{Z}^+$. Is it possible to prove that, for some given $m \in \{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23\}$, there are only finitely many $k$ such that the closed ...
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0 votes
0 answers
85 views

What will be the set of non-Wieferich numbers if the set of non-Wieferich primes is finite?

Integer $n$ is Wieferich number if $2^{\phi(n)}-1 \equiv 0 \pmod {n^2}$. Wieferich prime is Wieferich number with $n$ prime. It is an open problem if there are infinitely many Wieferich primes and ...
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5 votes
2 answers
190 views

Is it still not known whether the construction of shortest nonzero vector of a lattice w.r.t. $l^2$-norm is NP-hard?

It was shown in P. van Emde Boas, Another NP-complete partition problem and the complexity of computing short vectors in a lattice that the construction of a shortest nonzero vector of a Euclidean ...
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1 vote
1 answer
434 views

Finding an optimal covering trail for the set $\{0,1,2,3\}\times\{0,1,2,3\}\times\{0,1,2,3\}$

Here is a key question (i.e., Question 2 below) that, if correctly answered, would let me support a very general conjecture on a wide class of related problems, a conjecture that I have never shared ...
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1 vote
0 answers
24 views

Approximating spectra of (finite rank pertubations of) Laurent operators by spectra of (pertubations of) periodic finite operators

A tridiagonal matrix is a matrix which only has elements on three diagonals. So for $\alpha, \beta, \gamma \in \mathbb{C}$ consider the bi-infinite tridiagonal Laurent operator $T$ with $\beta $ on ...
6 votes
1 answer
314 views

Open problems in type theory

I am only a beginner in the field of type theory, and I'm wondering if the community could point me out a few open problems in the field. I have a good background in logic, in particular, proof theory ...
7 votes
4 answers
1k views

Are there any open sourced or crowd source math research projects?

I’m looking for open source math research projects as I know computer science students can easily have access to material they can contribute to but this is not the case with math as far as I’m aware. ...
6 votes
1 answer
281 views

What is the current status on bad tangent cones at isolated singularities?

Let $M^8 \subset B^9 \setminus \{ 0 \} \subset \mathbf{R}^9$ be a properly embedded, stable minimal hypersurface. Suppose that $0 \in \overline{M}$ is an isolated singularity of the surface. Question. ...
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2 votes
0 answers
207 views

Resources where one can find open problems in Analytic Number Theory

Edit: I have asked this question separately because I am in need of some resources which focus on analytic number theory only.So, this question is not a duplicate of:Resources where I can find open ...
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5 votes
0 answers
264 views

The "Smallest" open Diophantine Equation, a potential approach

In this question, Bogdan Grechuk found the "Smallest" open Diophantine equation, where size is determined by taking absolute values of all coefficients, then substituting 2 into all ...
3 votes
1 answer
80 views

Convex polygon shadows: Shortest equivalent segments

Let $P$ be a convex polygon. Q1. What is the shortest collection of line segments $S$ inside $P$ with the property that both $P$ and $S$ have the same sequence of orthogonal shadows as $P$ and $S$ ...
18 votes
2 answers
2k views

Mark Hovey's open problems in the theory of model categories

Mark Hovey maintains a list of open problems in model category theory. I think this list is quite old, and I don't know if Hovey is still updating it or not. My question is: i) which of the 13 ...
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1 vote
0 answers
231 views

Source for conjectures in commutative algebra

Do you know some books/survey papers/ websites on conjectures or open problems in commutative ring theory? I want to see if there are very famous open problems or conjectures in commutative ring ...
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0 votes
0 answers
53 views

Examples of a more general proof being easier to find than the proof itself? [duplicate]

Studying an open problem, it has become apparent that generalising and proving a more extensive theorem likely makes the problem simpler. Are there any well-known theorems where the author's ...
15 votes
5 answers
2k views

Resources where I can find open problems in number theory along with their level of difficulty

NOTE: I will not accept an answer because a lot of answers are really good and if anyone want to post under this question later then they are most welcome to post as comment or answer because it will ...
1 vote
0 answers
194 views

Is there a known connection between Wieferich primes and the Goormaghtigh conjecture?

I posted this question on SE, and was told I should repost it here. The Goormaghtigh conjecture explores the Diophantine equation of the form $$ \frac{a^b-1}{a-1}=\frac{c^d-1}{c-1}, $$ where $a>c&...
3 votes
2 answers
1k 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 ...
3 votes
1 answer
161 views

Status of finite-dimensional Ando's theorem

Szőkefalvi-Nagy's theorem says the following: if $A$ is a contraction on a Hilbert space $H$, then there exists a unitary $U$ on a Hilbert space $H'\supset H$ for which $A^n=P_HU^nP_H$ for all $0\le n$...
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0 votes
1 answer
490 views

Polynomials, $3^x$ and the Collatz conjecture

$\DeclareMathOperator\Orb{Orb}\newcommand\abs[1]{\lvert#1\rvert}$The Collatz or the $3n+1$ conjecture is open. Are there non-trivial polynomials $f(x)\in\mathbb Z[x]$ and $g(x)\in\mathbb R[x]$ having ...
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6 votes
0 answers
782 views

Is Collatz conjecture known to be true for specific 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 $3n+1$...
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2 votes
3 answers
414 views

On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part II

(Preamble: We have asked this same question in MSE two weeks ago, without getting any answers. We have therefore cross-posted it to MO, hoping that it gets answered here.) The topic of odd perfect ...
14 votes
2 answers
800 views

Open problems in symbolic dynamics

I would like to know which are some noticeable open problems in symbolic dynamics, including substitution dynamics. I'm especially interested in connections with topological chaos of various forms. ...
3 votes
1 answer
247 views

Planar subsets with many pairs of points on distance $1$ [duplicate]

Let $X$ be a subset of $\mathbb R^2$ consisting of $n$ distinct points. Let $d_1(X)$ be the number of pairs of points of $X$ on distance $1$ from each other. Define $$d_1(n)=\sup_{X\subset \mathbb R^2|...
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2 votes
0 answers
98 views

Inscribed square and convexity

Let $b(X)$ be the boundary of any $X$ subset of the plane. Does there exist $A,B$ convex compact sets of the plane, such that $C:=A\setminus B$ is simply connected and not empty, and such that ...
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4 votes
0 answers
191 views

Model theory and dynamical system (open problems)

I am curious about the open problems which are between model theory and dynamical system. I mean the open problems that are interesting for both groups and there are some evidences showing there might ...
1 vote
1 answer
230 views

Simple example of Hammerstein integral equation

I'm currently reading this paper (and working on a similar one). The main goal is to study the Hammerstein integral equation (in $\mathcal{C}(I,E))$: $$x(t) = \int_{0}^{t} K(t,s)f\big(s,x(s)\big)ds,\...
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3 votes
1 answer
452 views

Must Mersenne numbers be divisible by arbitrary large primes with exponent one?

Let $M_n$ denote the Mersenne numbers $M_n=2^n-1$. As $n$ varies, must $M_n$ be divisible by arbitrary large prime $p$ with exponent one, i.e. $p \mid M_n, p^2 \nmid M_n$? In other words, must the ...
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6 votes
0 answers
122 views

Is the unit-stick number of a knot equal to its stick number?

Define the unit-stick number $\sigma_1(K)$ of a knot $K$ to be the fewest unit-length sticks that can realize $K$. Clearly $\sigma_1(K)$ is at least the stick number $\sigma(K)$. It is known that the ...
24 votes
0 answers
905 views

0's in 815915283247897734345611269596115894272000000000

Is 40 the largest number for which all the 0 digits in the decimal form of $n!$ come at the end? Motivation: My son considered learning all digits of 40! for my birthday. I told him that the best way ...
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2 votes
0 answers
170 views

Factoring integers of the form $n=p q^2$ using elliptic curves

We got argument and strong experimental support that integers of the form $n=p q^2$ can be factored using elliptic curves easier than general integers Q1 Is this known? Added This is known since at ...
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1 vote
1 answer
331 views

Cramer–Castillon problem like

Special case of Golden ratio as a property of conic section (is it known?) as follows: Let $ABC$ be arbitrary triangle and $DEF$ is the its tangential triangle. Let $CF$ meets $AB$ at $G$ and $BE$ ...
13 votes
1 answer
1k views

Normal numbers, Liouville function, and the Riemann Hypothesis

This is a question about whether or not some number $\lambda^*$ is normal in base 2. More specifically, I am wondering if $\lambda^*$ is not normal. Proving it is normal would be next to impossible, ...
3 votes
0 answers
310 views

Dense sets in $\Bbb{R}^2$ with rational distance

We call a subset $S\subset \Bbb{R}^2$ rationally distanced if all $s_1,s_2 \in S$ have rational Euclidean distance. The Erdos-Ulam conjecture asks if there is a dense subset of $\Bbb{R}^2$ which is ...
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4 votes
0 answers
476 views

On RH in the Clay Institute list

As everybody knows, the Riemann Hypothesis is one of the problems of the millenium raised by the Clay Institute. Looking at the "official formulation" of various problems, say for instance ...
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0 votes
0 answers
116 views

How small the radical of $xyz(x+y+z)$ can be infinitely?

This is an open problem. Let $x,y,z$ be coprime integers (not necessarily pairwise coprime) and no proper subset sum of $\{x,y,z,-(x+y+z)\}$ is zero. For a quadruple $(x,y,z,-(x+y+z))$ define the ...
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3 votes
2 answers
475 views

Open problems in matroid theory

I read Oxley's book on matroid theory and found the theory fascinating. At the end, Oxley stated some open problems and conjectures in matroid theory. Are there any modern lists about such problems? ...
3 votes
0 answers
287 views

How to put a monetary incentive on an open research problem? [closed]

Since the old days, many famous mathematicians have been attaching monetary rewards to problems. I would like to put monetary bounties on some research mathematics problems. Some of those problems ...
7 votes
2 answers
955 views

Profound but not popular mathematical topics and notions

The algebraic Theory of Invariants used to be a hot topic until David Hilbert proved his two theorems about invariants. Then for tens of years, the popularity of the topic went down a long time before ...
5 votes
1 answer
273 views

Updated bounds or references for an old Erdős problem –– coloring the plane with multiple forbidden distances?

Define a graph $G_1$ where the vertices of $G_1$ are the points of the plane $\mathbb{R}^2$, and a pair of vertices $p, q$ is connected by an edge if and only if the Euclidean distance $d(p,q) =1$. ...
2 votes
0 answers
104 views

Two more counterexamples to a conjecture from 1975 about hamiltonicity of digraphs

Question from 2013 gives one counterexample to Nash-Williams's conjecture 1975 about hamiltonicity of dense digraphs. In the linked answer, @LouisD "reverse engineered" the counterexample ...
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5 votes
2 answers
711 views

Foundational results dependent on/equivalent to the continuum hypothesis or its negation?

I remember at a certain point early in my mathematical studies learning that the Axiom of Choice is equivalent to the following statement on Cartesian products: If $\{ X_i \}_{i \in I}$ is any ...
5 votes
0 answers
99 views

Frechet-Urysohn quotient of second countable locally compact Hausdorff space

In this paper from 2010 https://cmuc.karlin.mff.cuni.cz/pdf/cmuc1001/arhangav.pdf Arhangelskii asks if there is a quotient of a second countable locally compact Hausdorff space which is Frechet-...
5 votes
0 answers
258 views

Heuristic for a density conjecture related to the Collatz $(3x+1)$-problem

First, some notation. Define $T(n)$ over $n\in \mathbb{N}$ as: $$ T(n) = \left\{ \begin{array}{} 3n+1, & \text{if $n$ is odd}\ \\ n/2, & \text{if $n$ is even} \end{array} \right. $$ ...
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9 votes
2 answers
300 views

Generalized figures of constant width

Is it known which plane figures $Q$ can rotate touching three given circles $A$, $B$, and $C$? This question was asked by Lazar Lyusternik in 1946, there is only one reference to this paper that ...
-2 votes
2 answers
289 views

Basic research problems references [closed]

I have been looking for research problems in pure mathematics that I can try to solve for publishing papers. I am quite aware that it takes a lot of time and effort to get to a level where I can do ...
11 votes
2 answers
637 views

Which curves and surfaces are realizable by linkages? references?

Ok, so I try to formulate rigorously the question in the title, for which I am asking for references. My definitions may be flawed, so feel free to adjust/correct them! I care about dimensions 2 and 3 ...
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3 votes
0 answers
84 views

Effective radius of section of a convex set compared to that of the convex itself

The effective radius $er(A)$ of a $n$-solid $A$, is defined by Schramm (see the question by Gil Kalai Volumes of Sets of Constant Width in High Dimensions) to be the radius of the $n$-ball that has ...
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