2
votes
1answer
49 views

A possible minimal aperiodic set of corner Wang Tile

From one of my previous question Aperiodic set of corner Wang Tile (although it is put on hold), I realize there is a systematic way to construct aperiodic corner type of Wang tile from edge type ...
3
votes
1answer
164 views

Real analytic ergodic diffeomorphisms of the two sphere

Does there exists a real analytic area preserving ergodic diffeomorphism on $S^2$? (Possibly by perturbing a rotation in the real-analytic topology?)
7
votes
1answer
474 views

Ramanujan's problem 754 still open?

In addition to the MO question The Ramanujan Problems. , I would like to ask the following. Problem 754 from the list of the Ramanujan's problems ( ...
14
votes
0answers
422 views

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? ...
4
votes
0answers
195 views

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. ...
6
votes
4answers
1k views

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 ...
7
votes
4answers
453 views

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 ...
6
votes
1answer
549 views

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 ...
2
votes
1answer
210 views

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 ...
10
votes
1answer
464 views

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. ...
1
vote
0answers
239 views

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, ...
1
vote
1answer
517 views

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 ...
11
votes
7answers
1k views

Open problems in the theory of compact quantum groups

What are the important open problems in the theory of compact quantum groups? Or conjectures? Here is an example from An De Rijdt's Ph.D. thesis: Is every compact quantum group with the fusion rules ...
18
votes
0answers
430 views

The oriented homeomorphism problem for Haken 3-manifolds

Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But ...
9
votes
1answer
643 views

The Class Number One Problem for Real Quadratic Fields

An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick ...
37
votes
30answers
4k views

Fundamental problems whose solution seems completely out of reach [closed]

In many areas of mathematics there are fundamental problems that are embarrasingly natural or simple to state, but whose solution seem so out of reach that they are barely mentioned in the literature ...
3
votes
0answers
156 views

What is the status of the subadditivity problem for analytic capacity?

Hi, Here is another question that concerns analytic capacity. For a compact set $K$ in the plane, define the analytic capacity of $K$ by $$\gamma(K):=\sup|f'(\infty)|,$$ where the supremum is taken ...
1
vote
0answers
393 views

Reference Request - Jakob Weisblat's “The Search for the Odd Perfect Number” [closed]

Hi All! I am currently trying to locate an online copy of Jakob Weisblat's paper titled "The Search for the Odd Perfect Number". I could only get hold of the abstract: "A perfect number is a number ...
2
votes
0answers
163 views

What is the maximal density vertex subset of the 8-connected grid with induced vertex degrees $\leq 4$?

Let $G$ be the infinite graph defined by 8-connection of $\mathbb{Z}^2$. What is the maximal density vertex subset whose induced subgraph has maximum degree $\leq 4$? More precisely, what is the ...
29
votes
0answers
1k views

Shortest closed curve to inspect a sphere

Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and exterior to $S$ which has the property that every point $x$ on $S$ is visible to some point $y$ of ...
4
votes
1answer
748 views

Unsolved problems concerning Artinian Rings and Artinian Modules

I am preparing a write-up on Topics on Artinian Rings and Modules for a project. I hope to mention some unsolved problems in the domain of these objects along the way. Till now, I have been able to ...
21
votes
3answers
4k views

Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?

The Diophantine equation $$x^3+y^3+z^3=3$$ has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
0
votes
1answer
370 views

Existence of Solutions to an Equation Involving the Sum-of-Divisors Function [Reference Request]

Let $\sigma(x) = \sigma_1(x)$ denote the sum of all the positive divisors of $x$. If $n \in \mathbb{N}$ is odd and $\gcd(n, \sigma(n)) = 1$, then do there exist any solutions to the following ...
2
votes
1answer
511 views

Reference Request - Sharp Estimates for a Logarithmic Sum

Can anybody suggest a good (e.g. "non-technical") introduction to estimating bounds for logarithmic sums of the form $$\sum_{i=1}^{r}{{\alpha_i}{\log(q_i)}}$$ where the $$\alpha_i$$ are positive ...
4
votes
1answer
602 views

Reconstruction Conjecture: Group theoretic formulation

As we read from wiki, informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs. Is there a group-theoretic formulation of this ...
8
votes
1answer
1k views

Smallest dilation of a quadrilateral?

What is the smallest dilation of a quadrilateral in $\mathbb{R}^d$? This may be an open problem, which I know is verboten on MO. So my question is: Is this indeed open? It will take me some time to ...