# 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 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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### 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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### 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 ...
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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 ...
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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 ...
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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, ...
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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 f(n):=\begin{cases} n/2 & \text{if $n$ is even}\\ 5n+1 & \...
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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,... • 155 0 votes 0 answers 161 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 ... • 121 2 votes 0 answers 84 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 ... • 2,673 2 votes 2 answers 142 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 ... • 1,735 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 (... • 29 8 votes 0 answers 330 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 ... • 181 2 votes 0 answers 281 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 ... • 292 4 votes 0 answers 165 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? ... • 2,551 2 votes 0 answers 108 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 ... 11 votes 1 answer 722 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 410 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 ... • 121 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\... • 6,041 2 votes 1 answer 330 views ### Question about Jacobian conjecture on the reals By the works of Michiel de Bondt and Arno van den Essen, and Ludwik Drużkowski, it is known that if F=I+N, where I is the identity mapping and N is cubic homogeneous polynomials in n complex ... • 493 3 votes 1 answer 391 views ### \lim_{b \rightarrow \infty} {^{b}a} \in \mathbb{Q}_p for any a \in \mathbb{Z}^+? \newcommand\tetra[2]{{^{#1}{#2}}}In a recent discussion on the Tetration Forum (see https://math.eretrandre.org/tetrationforum/showthread.php?tid=1703&page=2), it has been pointed out how my ... • 1,100 2 votes 0 answers 199 views ### Is the Goldbach conjecture easier if we allow 1 as a prime? I hope this is the right site for the question. Is the Goldbach conjecture easier if we allow 1 as a prime? (12=1+11 would be allowed as Goldbach sum for 12) IOW: if we can prove Goldbach for the case ... • 129 35 votes 9 answers 4k 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 ... 8 votes 0 answers 495 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 115 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 180 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 ... • 1,100 0 votes 0 answers 99 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 ... • 24.2k 5 votes 2 answers 238 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 ... • 425 1 vote 1 answer 527 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 ... • 1,100 1 vote 0 answers 35 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 ... 7 votes 1 answer 613 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 2k 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 474 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. ... • 4,922 2 votes 0 answers 256 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 ... • 668 5 votes 0 answers 299 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 ... • 371 3 votes 1 answer 88 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 ... • 149k 20 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 ... • 1,474 1 vote 0 answers 380 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 ... • 1,347 16 votes 5 answers 3k 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 227 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&... 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 ... 4 votes 1 answer 188 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... • 465 1 vote 1 answer 559 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 ... • 13.6k 7 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... • 13.6k 2 votes 2 answers 473 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 ... 0 votes 0 answers 106 views ### 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 ... 14 votes 2 answers 888 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 296 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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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 ...