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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On improving the upper bound $I(m^2) \leq \frac{2p}{p+1}$, if $p^k m^2$ is an odd perfect number with special prime $p$
(Preamble: This question is an offshoot of this answer to an MSE question with the same title.)
Denote the classical sum of the divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$ and the ...
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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. ...
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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
$$d_1(n)=\sup_{X\subset \mathbb R^2|...
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Inscribed square and convexity
Let $b(X)$ be the boundary of any $X$ subset of the plane.
Does there exist $A,B$ convex compact sets of the plane, such that $C:=A\setminus B$ is simply connected and not empty, and such that ...
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Model theory and dynamical system (open problems)
I am curious about the open problems which are between model theory and dynamical system. I mean the open problems that are interesting for both groups and there are some evidences showing there might ...
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Simple example of Hammerstein integral equation
I'm currently reading this paper (and working on a similar one). The main goal is to study the Hammerstein integral equation (in $\mathcal{C}(I,E))$:
$$x(t) = \int_{0}^{t} K(t,s)f\big(s,x(s)\big)ds,\...
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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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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 ...
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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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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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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 $BE$ ...
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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, ...
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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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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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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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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?
...
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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 ...
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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 ...
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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 $d(p,q) =1$. ...
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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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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 ...
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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-...
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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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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 ...
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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 ...
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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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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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List of long open, elementary problems which are computational in nature
I would like to ask a question of a similar vein to this question.
Question: I'm asking for a list of long open problems which are computational in nature which a beginning graduate student can ...
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Isometries on the unit sphere
Suppose that $X$ and $Y$ are two Banach spaces, $S_{X}$ and $S_{Y}$ their unit spheres, and $f$ an onto isometry between $S_X$ and $S_Y$. Does it follow that $X$ and $Y$ are isometric?
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Has Pillai conjecture been proven?
I found the paper https://hal.archives-ouvertes.fr/hal-00698687v9/document which claims the proof of Pillai conjecture.
However, it is not mentioned anywhere that it has been proved. It's stated ...
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Conjectures and open problems in representation theory [closed]
Are there very famous open problems or conjectures in representation theory, or in enumerative geometry, like the volume conjecture in topology?
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Updates to Stanley's 1999 survey of positivity problems in algebraic combinatorics?
[I am a co-moderator of the recently started Open Problems in Algebraic Combinatorics blog and as a result starting doing some searching for existing surveys of open problems in algebraic ...
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When do two knots have isomorphic fundamental bikeis?
A kei, also known as an involutive (or involutory) quandle, is a quandle $(Q,*)$ satisfying the involution condition that $(x*y)*y=x$ for all $x$ and $y$. Just like we can define a fundamental ...
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Is every finite-order unimodular matrix conjugate to a $0,1,-1$ matrix?
Problem. Given a matrix $A\in\mathrm{GL}(n,\mathbb{Z})$ such that $A^k=1$ for some $k\geq 1$, is there a matrix $g\in\mathrm{GL}(n,\mathbb{Z})$ such that $gAg^{-1}$ has only $0$, $1$, and $-1$ as ...
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Is it an open problem whether fast-growing hierarchies can be defined without fundamental sequences?
Googology Wiki says this, concerning the relation between fast-growing hierarchies defined for all countable ordinals, and the existence of a system of assigning a canonical fundamental sequence to ...
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What's an example of an $\infty$-topos not equivalent to sheaves on a Grothendieck site?
My question is as in the title:
Does anyone have an example (supposing one exists) of an
$\infty$-topos which is known not to be equivalent to sheaves on a
Grothendieck site?
An $\infty$-topos is as ...
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Decision problems for which it is unknown whether they are decidable
In computability theory, what are examples of decision problems of which it is not known whether they are decidable?
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A question regarding Kadison-Kaplansky idempotent conjecture (A nearest group element $g$ to a nontrivial self adjoint unitary element u )
Edit: According to answer and comments by Prof. Valette we edite the question.
The Kadison Kaplansky conjecture says:
Kadison-Kaplansky conjecture: If $G$ is a torsion-free discrete group then $C^*_{\...
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What is the state of research on finding all prime knots with 17 crossings?
In this 1998 journal paper, all the prime knots with 16 or fewer crossings are found (some of which were found earlier by others). There are over 1.7 million such knots. But the prime knots with 17 ...
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Which knot invariants have no known diagram-independent descriptions?
Many knot invariants in knot theory are discovered by finding a property of knot diagrams which is invariant under the three Reidemeister moves. Now in principle, any knot invariant can be described ...
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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 $Q(x)$ ...
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The Total Graph is similar to a line graph
Consider the total graph of a regular graph. From the structure, it seems that it has a similar structure to the line graph ( two different sub-cliques joining at a single point) except that, in ...
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How many solutions of x^3 +y^3 = z^3+3 are known? [duplicate]
I've been told that there is reason to think that the equation
$x^3 + y^3 = z^3 + 3$ has solutions in positive integers other than
$$4^3 + 4^3 = 5^3 +3.$$
Can someone tell me the current status of ...
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A cohomology associated to a (not necessarily integrable) distribution (Hilbert 16th problem and dynamical Lefschetz trace formula 2)
Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{|...
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Hyperbolic groups and spaces of negative curvature
Mikhail Gromov states that he "tried for about 10 years to prove that every hyperbolic group is realizable by a space of negative curvature" in his interview with Martin Raussen and Christian Skau (...
13
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Hilbert 16th problem and dynamical Lefschetz trace formula
I would like to apply the known version of the conjectural formula (11) page !0 of the paper Number theory and dynamical Lefschetz trace formula.
Disclaimer: I do not have a complete ...
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Coloring a graph formed by cliques sharing at most one point
Consider a graph formed by $k$ $k$ order cliques sharing at most one point. Consider thedegenerate case of all cliques disjoint, which is trivially $k$ colorable. Now, to colour any other such graph, ...
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Is the exact solution of the wave equation for the scattering of waves by a disk/cylinder an open problem?
The solution exact solution of the Helmholtz equation for the scattering of waves by a sphere is relatively straightforward and has been known since the time of Lord Rayleigh.
The exact solution of ...
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Open problems concerning all the finite groups
What are the open problems concerning all the finite groups?
The references will be appreciated. Here are two examples:
Aschbacher-Guralnick conjecture (AG1984 p.447): the number of conjugacy ...
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Open problems in Sobolev spaces
What are the open problems in the theory of Sobolev spaces?
I would like to see problems that are yes or no only. Also I would like to see problems with the statements that are short and easy to ...