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?"
591 questions
4
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Is there a natural topos where the Riemann hypothesis is provable or disprovable?
While constructive logic is compatible with classical logic and is sufficient to develop almost all important theorems from classical complex analysis, constructive is also compatible with axioms that ...
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10
votes
1
answer
586
views
No real roots of $\frac{1}{2^{1+it}} + \frac{1}{3^{1+it}} + \frac{1}{5^{1+it}}$
Prove or disprove that $$\frac{1}{2^{1+it}} + \frac{1}{3^{1+it}} + \frac{1}{5^{1+it}} \neq 0$$ for any real number $t$.
I find it surprising that so simple looking equations involving complex numbers ...
- 163
7
votes
0
answers
280
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Polynomial identification of natural numbers
Let $n$ be an integer parameter. Is there a polynomial $f(x,y,n)\in \mathbb{Z}[x,y,n]$ such that
$$
n\in \mathbb{N} \Longleftrightarrow \exists x,y\in \mathbb{Z}, f(x,y,n)=0?
$$
This is a generalized ...
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10
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0
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468
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An inequality that implies Frankl's union-closed conjecture
Let $\mathcal{F}$ be an union-closed family of subsets of $[n]=\{1,2,...,n\}$, assume $\varnothing\in\mathcal{F}$. Let $l_i=|\{S|S\in\mathcal{F},i\notin S\}|,u_i=|\{S|S\in\mathcal{F},i\in S\}|$, then $...
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5
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What are some open problems in geometric probability?
What are some open problems in geometric probability?
Context: My question, "A tetrahedron's vertices are random points on a sphere. What is the probability that the tetrahedron's four faces are ...
- 5,059
1
vote
0
answers
120
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Existence of n-Level Decomposition Trees for Large Primes
I am investigating a combinatorial structure on prime numbers called an n-Level Decomposition Tree (DT$_n$). The definition is as follows:
Let $p > 17$ be a prime number. An n-level decomposition ...
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40
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10
answers
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Which pairs of mutually contradicting conjectures are there?
Years ago I had the pleasure of witnessing Simon Thomas giving a wonderful talk about Martin's conjecture, which I just now fondly remembered reading this question. Even though I am not well-versed in ...
7
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0
answers
656
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Is Landau's 4th problem the smallest unsolved problem in number theory?
The 4$^{th}$ of https://en.wikipedia.org/wiki/Landau%27s_problems is on the infinity of primes that are one more than a square.
The answer depends on certain assumptions about allowable statements and ...
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2
votes
1
answer
361
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An approach to a generalization of Frankl's union-closed sets conjecture
Let $I$ be a non-empty finite set, $\mathcal{F}$ be a non-empty union-closed family of subsets of $I$ except the empty set and $n$ real numbers $x_i\geq1,i\in I$. Let $d_i=\sum_{i\in J,J\in\mathcal{F}}...
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13
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1
answer
437
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Pillai's arithmetical function and primality
Pillai's arithmetical function is defined by $P(n)=\sum_{k=1}^n\gcd(k,n)$.
Is it true that $1+P(n)\equiv 0\pmod{n}$ iff $n$ is prime?
It was stated without proof in 2014 on the OEIS A018804 and has ...
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0
votes
0
answers
103
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Irrationally drifting Schwarz reflections
Let
$$ \alpha \in (0, \pi), \qquad \frac{\alpha}{\pi} \notin \mathbb{Q}, $$
and fix a bounded sequence of “drift-points”
$$ (c_k)_{k \in \mathbb{Z}} \subset \mathbb{C} $$
whose set of accumulation ...
0
votes
0
answers
110
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Tilted BBM Spines → Brownian Net?
Let $\mathcal{B}$ be critical binary branching Brownian motion on $\mathbb{R}$ started from a single particle at the origin. Fix $k \ge 1$ and a constant $\lambda > 0$. Condition on the atypical ...
3
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0
answers
178
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Can a uniformly-bounded Klein–Gordon wave sweep its nodal set across a rigid obstacle?
Consider Minkowski space $\mathbb R^{3+1}$ and the massive Klein–Gordon equation
$$ \square\varphi + m^{2}\varphi = 0 . $$
Let $\Omega\subset\mathbb R^{3}$ be a fixed smooth bounded “obstacle’’ with ...
0
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1
answer
258
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Furstenberg's topological proof method (of prime infinitude) might be more powerful than initially thought. $\sigma_0(\frac{p_n - 1}{4})$ odd i.o.?
Let $\sigma_0(n)$ be from number theory, i.e. the total number of divisors of an integer $n$. Let $p_n$ denote the $n$th prime number.
Let $S =$ the set of prime numbers $p \in 4\Bbb{Z} + 1$ such ...
1
vote
0
answers
382
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More about the algebraic strengthening of Frankl's union-closed conjecture
Continue my previous question, consider the first conjecture:
Let $\mathcal{F}$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$.
Conjecture: ...
- 1,558
3
votes
0
answers
411
views
A formula for minimum number of $k$-element nonnegative-sum subsets of a zero-sum set: Revisiting Manickam-Miklós-Singhi conjecture
Define the quantity $A(n,k)$ as follows:
$A(n,k)$ denotes the minimum number of $k$-element nonnegative-sum subsets of $n$ arbitrary real numbers $x_1,\dots,x_n$ whose sum is non-negative.
More ...
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1
vote
1
answer
297
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Reference request: Open problems about finite free products of finite groups
I'm working with finite free products of finite groups, i.e. a group $G$ given by $$G = F_1 \ast \ldots \ast F_n$$ where each $F_i$ is finite.
Do you know of any open problems as well as references ...
11
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1
answer
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Why is Erdős' conjecture on arithmetic progressions not discussed much, and is there an active pathway to its resolution?
Erdős' conjecture on arithmetic progressions (also known as the Erdős–Turán conjecture) is a major open problem in arithmetic combinatorics. It asserts that if the sum of the reciprocals of the ...
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5
votes
1
answer
647
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Updated versions of problems in geometric group theory
Is there an updated version of Questions in geometric group theory, by M. Bestvina or Some group theory problems, by M. Sapir? What are the recent problems in geometric group theory?
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0
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The behaviour of $\mathsf{HOD}^{L[x]}$ on a cone
In "Introduction to $Q$-Theory" by Kechris-Martin-Solovay in the proceedings of the Cabal Seminar 1979 - 1981, back when it was believed axioms up to $I_3$ could be compatible with $\Delta^...
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13
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1
answer
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Collaborative repositories on open problems? [closed]
When proving a mathematical statement (or solving any problem, not necessarily mathematical), usually one abstracts the concepts within the statement/problem, and "searches" for theorems / ...
9
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2
answers
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Is it possible to discover a "Universal formula" that generates and generalizes all odd Collatz numbers?
I am just an ordinary student, and I have never had the chance to ask this question to mathematicians. Perhaps this is the first opportunity I have to ask it here.
When we refer to "Collatz ...
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votes
4
answers
2k
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Elementary consequences of famous technical theorems and/or conjectures
Recently, Will Sawin gave a perfect answer to my question about elementary consequences of Langlands program. Then Timothy Chow asked a similar question about non-abelian class field theory, and ...
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6
votes
1
answer
344
views
Conjecture closed form of summation over an integer lattice
Conjecture:
$$\forall \Lambda,\ \exists P(x)\in \mathbb{Z}[x],\ S(\Lambda):=\sum_{k\in\Lambda}\prod_{j=1}^{n}\frac{1}{1+k_{j}^2}=\frac{\pi^{n}}{\sinh^{n}(\pi)\operatorname{d}(\Lambda)}P\left(\cosh\...
user500489
2
votes
1
answer
325
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Maximum number of connected components of surfaces in three dimensions, what is known?
Part of Hilbert's 16th problem is:
It seems to me that a thorough investigation of the relative positions of the upper bound for separate branches is of great interest, and similarly the ...
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0
answers
452
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What is the status of this conjecture on symplectic forms "standard-at-infinity" on $\mathbb{R}^{2n}$?
In McDuff and Salamon's Introduction to Symplectic Topology, the following open problem is mentioned.
Problem 50 (Standard-at-infinity)
Let $n \ge 3$ and let $\omega$ be a symplectic form on $\mathbb{...
12
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0
answers
598
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Strengthening of Frankl's union-closed sets conjecture: An algebraic approach
Let $\mathcal F$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$.
Conjecture: There exists $k\in [n]$ such that:
$$\sum_{k\in A,A\in \mathcal F}\...
- 1,558
2
votes
0
answers
96
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Have the open Questions 1 and 2 from Section 7 of the paper "Integrals with values in Banach spaces" been answered?
Some context: I had previously asked the post below on MSE, but someone suggested I ask it here and delete the original post.
In section 7 of the paper Integrals with values in Banach Spaces and ...
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votes
9
answers
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What is the oldest open math problem outside of number theory?
The question of "What is the oldest open problem in mathematics?" comes up from time to time, and there seems to be consensus that the answer is "Are there any odd perfect numbers?"...
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7
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2
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Poisson binomial conjecture
Let $X_i\in\{0,1\}$
be mutually independent and distributed according to $\mathrm{Bernoulli}(p_i)$
and similarly, $Y_i\sim\mathrm{Bernoulli}(q_i)$,
for some parameters $p,q\in[0,1]^n$. Put $X:=\sum_{i=...
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vote
0
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293
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Solving functional analysis problems by using Algebraic geometry
I am thinking about some open problems in nonlinear functional analysis and I just wanted to know if there are any problems that have been solved by using Algebraic geometry techniques in these fields....
1
vote
0
answers
287
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Papers showing two open problems are connected [closed]
I'm currently writing an article in which I connect two open problems from different fields of mathematics (group theory and combinatorics). While writing the introduction, I was trying to think of ...
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2
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0
answers
291
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Open problems in differential algebra and affine algebraic geometry
I am currently doing a PhD in differential algebra and affine algebraic geometry at the University of Buenos Aires. I've been struggling to find a list of interesting and big open problems in these ...
4
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0
answers
244
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Frankl's conjecture for infinite lattices
Given a poset $L$, call it trivial if $\left|L\right| < 2$ and let $\mathcal I\left(L\right)$ be its poset of ideals, $\mathcal C\left(L\right)$ be its set of chains, and $\mathcal M\left(L\right)$ ...
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1
answer
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Open problems in complete theories
It is well-known that every complete recursively enumerable first-order theory is decidable. Does that mean that such theories are "trivial", or are there still interesting open problems ...
user507115
2
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0
answers
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Conjecture: $x^4+1$ is never Wieferich prime
Related to this question and Alexander Kalmynin's answer.
For natural $n$ define $J(n)=(2^{n-1}-1) \bmod n^2$
and if $n$ is power of two define $J(2^n)=1$ (this is artificial, just to
avoid triviality ...
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3
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1
answer
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Probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$
Using my computer, I found that in the interval $[1, N]$ the probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$ is greater than constant $c$ where $N=10^2, 10^3,...,10^{9}$, $x$ ...
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3
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No starter "accessible" well known open problems
We all know very famous open problems. Usually the ones that become famous are well studied and lots of progress was achieved and conjectures, partial results, reductions and so on exist. This is one ...
6
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1
answer
362
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Congruence obstructions for three consecutive powerful numbers
A powerful number is an integer $m$ such that if $p$ is prime and $p \mid m$ then $p^2 \mid m$.
Powerful numbers can be represented in the form $m=u^2 v^3$.
Erdos conjectured that three consecutive ...
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0
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1
answer
188
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Could I possibly exploit distinct odd primes raised to 6 to solve Exact Three Cover, when reducing it in Subset Sum?
I'm solving Exact 3 Cover, given a list with no duplicates $S$ of $3m$ whole numbers and a collection $C$ of subsets of $S$, each containing exactly three elements. The goal is to decide if there are $...
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0
answers
187
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About geodesic vector fields and the status of a classic problem on the number of closed geodesics
A classical problem in differential geometry is to determine whether every compact Riemannian manifold admits infinitely many geometrically distinct closed geodesics. A good reference on the subject ...
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6
votes
1
answer
266
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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 ...
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2
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0
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345
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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 ...
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29
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6
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3k
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Open problems which might benefit from computational experiments
Question: I wonder what are the open problems , where computational experiments might be helpful? (Setting some bounds, excluding some cases, shaping some expectations ).
Grant program: The context of ...
2
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1
answer
268
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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 ...
- 161
4
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0
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427
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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 ...
- 2,123
0
votes
1
answer
495
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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, ...
- 31
9
votes
2
answers
2k
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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 & \...
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7
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0
answers
217
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,...
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votes
0
answers
313
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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 $A_p$ ...
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