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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An open problem in convex geometry
Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...
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2
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Floors of rationals to powers: Infinite number of primes?
Let $r=a/b$ be a rational number in lowest terms, larger than $1$,
and not an integer (so $b > 1$).
Q. Does the sequence
$$ \lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor,
\...
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0
answers
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Stability of analytic Zariski structures
Noetherian Zariski structures are introduced by Hrushovski and Zilber(1996).
An analytic Zariski structure is a generalization of Noetherian Zariski structures, introduced by Zilber and Peatfield.
...
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answers
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Automorphisms of the L-function associated to an elliptic $\mathbb{Q}$-curve
$\DeclareMathOperator\Aut{Aut}\newcommand{\alg}{\mathrm{alg}}\newcommand{\an}{\mathrm{an}}$Edited after Noam Elkies' comment: From what I understand (very little actually), there exist elliptic curves ...
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Second duals of Grothendieck spaces
The classical example of a Grothendieck space is $\ell_\infty$. It is also known that its even duals $\ell_\infty^{**}$, $\ell_\infty^{(4)}$, $\dots$ are Grothendieck spaces.
(See, e.g., this note ...
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1
answer
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Positively curved Riemannian manifolds
Let $M$ be a compact Riemannian manifold with positive sectional curvature whose universal covering space is diffeomorphic to $S^n$. Is $M$ diffeomorphic to a spherical space form?
I know, by a ...
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4
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Interactions of number theoretic conjectures and other fields of mathematics
There are many interesting open conjectures in number theory. My question is not about partial results or possible ways to prove them. It is about their interactions with the other fields of ...
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1
answer
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Any results towards the irrationality of the sum of reciprocals of perfect numbers? [closed]
This question is a follow up to my comment to Sum of the reciprocal of perfect numbers. I would like to know which results have been published about the possible irrationality of the sum of ...
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1
answer
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A Question on 1, 2 ,3 Conjecture
The 1, 2, 3 conjecture is well-known:
If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the ...
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4
answers
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A Special Pair of Models for ZFC (New Version)
Are there two models $M$ and $N$ for $\text{ZFC}$ such that:
(1) $M\subseteq N$
(2) $\aleph_{1}^{N}=\aleph_{1}^{M}$
(3) $\aleph_{2}^{N}=\aleph_{\omega +1}^{M}$
Update: According to Peter's useful ...
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Forcing with c.c.c forcing notions, Cohen reals and Random reals
I think the following question is due to Prikry:
Question. Is it consistent that any non-trivial c.c.c forcing notion adds a Cohen real or a Random real?
Is the question still open? What partial ...
5
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1
answer
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A Hot Betting On HOD
Remark: This question is based on an open question at the end of a paper by Hamkins, Kirmayer, and Perlmutter: "Generalizations of the Kunen Inconistency".
$HOD$ as an inner model of $ZFC$ lies ...
4
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answers
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A challenging non homogenous fractional inequality
I have posted this question on Stackexchange but it has received no answer so far. It is a challenging generalization of several difficult inequalities, where none of the usual methods used in ...
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1
answer
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Euclidean real quadratic fields
It is known that, under GRH, a real quadratic field is Euclidean iff it is a UFD. So, assuming the conjecture of Gauss and GRH, we expect that there are infinitely many Euclidean real quadratic fields....
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Decidability of equality of elementary expressions
In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them.
Define the set $\mathcal{E}$ of ...
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2
answers
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Is the Steiner ratio Gilbert–Pollak conjecture still open?
Gilbert-Pollak conjecture on the Steiner ratio: Consider a set $P$ of $n$ points on the euclidean plane. A shortest
network interconnecting $P$ must be a tree, which is called a Steiner minimum ...
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Is $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$ correct, where $n$ is an integer?
Is it true that $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$, where $n$
runs over the integers?
The existence of the limes inferior follows from Dirichlet's approximation theorem,
but the ...
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4
answers
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Product of exponents of prime factorization
Let $p(n)$ be the product of the exponents of the prime factorization of $n$. For example,
$$p(5184) = p(2^6 3^4) = 24 \;,$$
$$p(65536) = p(2^{16}) = 16 \;.$$
Define $P(n)$ as the number of iterations ...
2
votes
0
answers
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What are the divisors of $2n^2 - \sigma_{1}(n^2)$ for composite $n$?
What are the divisors of $2n^2 - \sigma_{1}(n^2)$ for composite $n$?
Here, $\sigma_{1}$ is the classical sum-of-divisors function. For example, $\sigma_{1}(3^2) = 1 + 3 + {3^2} = 13$.
(The function ...
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0
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Can six square numbers be simultaneously represented in a single sum of consecutive odd numbers? [closed]
I had some free time from my work to do a little exploration regarding the existence (or non existence) of perfect cuboids. A solution is represented by the set of Diophantine equations:
$a^2 + b^2 = ...
7
votes
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answers
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Graphs with graphic imbalance sequences
Let $G$ be simple undirected graph and $e=uv\in E(G)$.
The imbalance of the edge $e$ is the value $imb(e)=|d(u)-d(v)|$.
Let $M_{G}$ denotes the imbalance sequence (or more correctly, multiset of ...
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answer
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Passing C through a slot
Question: Given a closed curve C, what will be the (bounds on) dimension of the interval it will pass through?
i.e. which are the necessary and sufficient conditions for a planar compact set C to pass ...
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Tarski Monster group with prime 5
Does the Tarski Monster group with prime 5 exist? I know that for 2 and 3, the group does not exist, but what about 5?
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0
answers
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Do the banded operators check the invariant subspace problem?
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
3
votes
1
answer
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Equiprojective polyhedra
Seeing Garabed Gulbenkian's question (which was inspired by Joel Hamkins' question), reminds me of an analogous problem which I believe remains open,
and which some might find intriguing.
Define an ...
16
votes
2
answers
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Tiling the square with rectangles of small diagonals
For a given integer $k\ge3$, tile the unit square with $k$ rectangles so that the longest of the rectangles' diagonals be as short as possible. Call such a tiling optimal. The solutions are obvious in ...
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votes
3
answers
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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.
...
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votes
1
answer
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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 ...
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answers
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Set-theoretic reformulation of the invariant subspace problem
The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and $l^2$...
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answer
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A question about the Axiom of Choice
Let AC denote the Axiom of Choice. Let PP denote the so-called "Partition Principle" which states that "If S is a non-empty set and T is a non-empty set of pairwise disjoint subsets
of S, then S can ...
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answers
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Open problems in algebraic topology and homotopy theory
Some time ago (I see it was initially written before 1999?) Mark Hovey assembled a list of open problems in algebraic topology. The list can be found here. Some of the problems I know about have been ...
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answer
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A conjecture by Euler about $8n+3$
Euler's conjecture: For any positive integer $n$, $8n+3$ can be represented as a sum
$$8n+3=(2k-1)^2+2p,$$
where $k$ is a positive integer, and $p$ is a prime.
I want to know whether there has been ...
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votes
0
answers
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Counting factors: is this approach in the literature on multiperfect numbers?
Does the following approach (or something near it) exist in the number theory
literature?
I will provide some motivation for $\omega(p^n - 1)$ as $n \rightarrow \infty$
and for this question. First, ...
5
votes
1
answer
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List of open problems of formal languages [closed]
As we know, there are some open problems of formal languages. I am wondering if there is a somehow complete list of open problem of formal languages. If there isn't such a list, can we make it one as ...
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Does this modified Hasse principle hold for curves?
Let $C$ be a curve over $\mathbb Q$ with a point $P$ on $Pic^1$. For each $\mathbb Q$-rational point $Q$, $Q-P$ is a point on the Jacobian $J$. We can use the map $H^0(\mathbb Q, J) \to H^1(\mathbb Q,...
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votes
1
answer
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Needle probing for a convex body
Suppose there is an unknown closed convex body $K$ of
volume vol$(K) = V$ inside the
unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$.
You are permitted to probe with a (one-dimensional)
...
12
votes
2
answers
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Distinctive property of the primes 17 and 19?
Consider the question whether it is true that a prime number $p$ divides
$1^1+2^2+3^3+....+(p-1)^{p-1}$ if and only if $p \in \{17,19\}$.
For the obvious heuristic reasons, for large $n$ one would ...
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votes
3
answers
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Optimal inspection path on a sphere
Suppose you would like to "inspect" every point of a unit-radius
sphere $S \subset \mathbb{R}^3$ by walking along a path $\gamma$
on $S$, but you can only see a distance $d$ from where you ...
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vote
2
answers
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A conjecture on closed discrete subset
I am struggling with this old problem, which is also posted here:
Let $X$ satisfy countable chain condition(abbreviated as CCC) and $X$ has a regular $G_\delta$-diagonal. Then the cardinality of $X$...
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votes
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answers
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Surfaces filled densely by a geodesic
Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no
single geodesic $\gamma$ that fills $S$ densely?
Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points
...
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votes
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answers
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Open problems in PDEs, dynamical systems, mathematical physics
(This question might not be appropriate for this site. If so, I apologize in advance. I would have posted to mathstack, but I'm looking for advice from active researchers.)
I am an undergrad in math ...
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Known and unknown about Ramanujan's tau function
What is a good reference for open problems relating to the Ramanujan tau function?
I know about Lehmer's conjecture. I know the following reductions of the problem: the smallest counterexample must ...
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votes
1
answer
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Collatz conjecture— finite state machine transducer construction, origination?
wikipedia has an entry on the Collatz conjecture with a section on As an abstract machine that computes in base two. this apparently describes a construction of a FSM transducer computing sequential ...
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1
answer
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Is it true that every f.g. infinite simple group has exponential growth?
Is it true that every finitely generated infinite simple group has
exponential (word-)growth?
Remark: As Mark Sapir has pointed out, the question whether
every finitely generated group of ...
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votes
4
answers
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problems from the scottish book
Which of the problems from the Scottish Book (pdf of English version) by Stefan Banach are still open? I know that one of the problems was solved by Per Enflo for which he got a live goose from ...
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votes
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answer
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Is there any o-minimal expansion of the real field with functions of growth higher than exponential?
Let $\bar{\mathbb{R}}$ be the structure of the real field, that is $(\mathbb{R},0,1,+,-,*,<)$ . We say that a function $f$ is of growth higher than exponential if for all $N\in \mathbb{N}$ there $f(...
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answer
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An open problem on general topology
There is an open problem in this paper: Classes defined by stars and neighbourhood assignments by van Mill and others.
Problem 4.8. Is a regular (Tychonoff) star compact space metrizable if it has a $...
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open problems in Seiberg-Witten Theory on 4-Manifolds
What are some of the open problems in Seiberg-Witten Theory on 4-Manifolds.I tried googling but couldn't any. I tried googling it, but couldn't find any resources.The places where I can a survey or ...
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How many values a polynomial map misses?
Let $F$ be a field. For a uni-variate polynomial $f(x)$ over $F$,let $M_f(F)$ denote the number of values that $f$ misses, that is, the cardinality of the subset $F - f(F)$ in $F$. Assume that $f$ is ...
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Randomly switching street lights, in a square city
This is a combinatorics-probability question, best stated however in "recreational" terms. Imagine a $N\times N$ city, meaning that we have $N$ horizontal streets, and $N$ vertical streets. At each ...