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 ...

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26
votes
1answer
552 views

A long-lasting conjecture of Pólya & Szegő

There is a conjecture by Pólya & Szegő (~1950) which is as follows: "Of all $n$-gons of a fixed area, the regular $n$-gon minimizes the first Dirichlet eigenvalue." Surprisingly, this is still ...
1
vote
1answer
132 views

On even almost perfect numbers other than powers of two

(Note: This question is an improved version of and has been cross-posted from this MSE post.) Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) = 2x - 1$, then we call $x$ an ...
13
votes
2answers
649 views

Torsion-freeness of two groups with 2 generators and 3 relators and Kaplansky Zero Divisor Conjecture

Let $G_1$ and $G_2$ be the groups with the following presentations: $$G_1=\langle a,b \;|\; (ab)^2=a^{-1}ba^{-1}, (a^{-1}ba^{-1})^2=b^{-2}a, (ba^{-1})^2=a^{-2}b^2 \rangle,$$ $$G_2=\langle a,b \;|\; ...
2
votes
1answer
223 views

A generalization of Frankl's conjecture?

Would it be reasonable to conjecture what follows : there is a real constant $c > 1/2$ such that, for every natural number $n$, if $X_{1}, \ldots , X_{n}$ is a union-stable family of distinct ...
15
votes
3answers
604 views

Open problems in Hopf algebras

I couldn't find a list of open problems in Hopf algebras. So my question is the following: In the theory of Hopf algebras, what are the (big) open problems? Any kind of problem/question will be ...
0
votes
0answers
41 views

If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n^2$ solitary?

(Note: A similar question was asked in MSE two months ago.) Let $\sigma(x)$ be the sum of the divisors of the natural number $x$, and denote the abundancy index $\sigma(x)/x$ by $I(x)$. Here is my ...
1
vote
1answer
334 views

If $N = qn^2$ is an odd perfect number, is it possible to have $q + 1 = \sigma(n)$?

The title says it all. Question If $N = qn^2$ is an odd perfect number with Euler prime $q$, is it possible to have $q + 1 = \sigma(n)$? Heuristic From the Descartes spoof, with quasi-Euler ...
1
vote
0answers
19 views

The complexity of Max-K interval selection

I came up with the following problem, but do not know how to analyze it. Let $S$ be an ordered set of integers with size $n$ (i.e., $S=\{1,2,...,n\}$). An interval $INV(a,b)$ covers the elements in ...
9
votes
1answer
340 views

A strengthening of Frankl's union-closed sets conjecture?

A Frankl family is a nonempty finite family $\mathcal F$ of nonempty finite sets such that $A,B\in\mathcal F\implies A\cup B\in\mathcal F.$ Define $d_\mathcal F(x)=|\{A\in\mathcal F:x\in A\}|$ and ...
30
votes
8answers
5k views

What are some important but still unsolved problems in mathematical logic?

In the past, First order logic and its completeness and whether arithmetic is complete was a major unsolved issues in logic . All of these problems were solved by Godel. Later on, independence of ...
5
votes
0answers
83 views

Does every model of $I\Delta_0$ has an end extension to a model of $I\Delta_0+\Omega_1$?

Does every model of $I\Delta_0$ has an end extension to a model of $I\Delta_0+\Omega_1$? In End extensions of models of linearly bounded arithmetic paper the author said this problem is open. I ...
8
votes
1answer
652 views

Some Questions on the Collatz conjecture

The set of all positive whole numbers is denoted by $\mathbb{N}_+$. Let $f\colon\ \mathbb{N}_+\to\mathbb{N}_+:n\mapsto \begin{cases}\frac{n}{2}&\text{$n$ even}\\3n+1&\text{$n$ ...
-4
votes
2answers
141 views

Diagonal argument for even perfect numbers

Following this, let's define the notion of perfect sequence as follows: $(u_{i})_{i}$ is a perfect sequence if and only if it is the sequence of divisors of an even perfect number in increasing order ...
12
votes
1answer
743 views

On cubic reciprocity for $x^3+y^3+z^3 = 996$?

I. The Diophantine equation, $$x^3+y^3+z^3 = 3w^3\tag1$$ with $x\geq y \geq z$ and $w=1$ has only two known solutions, namely $1,1,1$ and $4,4,-5$. Are there larger ones? As Noam Elkies points out ...
7
votes
1answer
297 views

What are some open problems regarding elliptic curves in finite fields?

I accept that my question seems so vague and broad, and I already looked into some similar questions in MO. But I would like to learn specifically about some open problems and conjectures regarding ...
1
vote
0answers
37 views

On a conjecture about Riemannian metric with positive sectional curvature [duplicate]

What is the last status of the following conjecture? Is it still open? What partial or similar results are known up to now? Conjecture: $S^2\times S^2$ admits a Riemannian metric with positive ...
18
votes
3answers
621 views

All compact surfaces $S\subseteq \mathbb{R}^3$ are rigid?

Recently I've come across a lecture in differential geometry by Fernando Codá (in portuguese!) in which he stated that the following problem is (at least, up to 2014) open: Given ...
1
vote
2answers
259 views

Looking for (information about) long diamonds

I was given an open problem as a birthday present recently. While I can probably handle spoilers at this point, what I really want are literature and other references. Also acceptable would be ...
4
votes
0answers
180 views

What is behind the Hodge conjecture? [duplicate]

My question is quite naive, and my knowledge limited on the subject. I heard lot of talks about Hodge conjecture. I wanted to ask about an intuitive way to figure out why we should care about Hodge ...
10
votes
0answers
301 views

Progress in Guy's “Unsolved problems in Number Theory”? [closed]

I often peruse through Guy's book whenever I'm not being boggled down by my research. It crossed my mind today if any of these "unsolved problems" have become indeed solved. I thought about doing a ...
5
votes
1answer
167 views

Does $ZF$+$LM$+$A_{\lt2^{\aleph_0}}$ imply $\lnot$$WCH$?

In what follows, I will use the following acronyms: $AS$:= Freiling's Axiom of Symmetry $LM$:="Every set of reals is Lebesgue measurable." $WCH$:="every uncountable subset of $\mathbf R$ can be put ...
11
votes
2answers
323 views

Gerstenhaber conjecture for free loop space

I- Is the following statement still a conjecture see this article ? Conjecture (?) Let $M$ be a simply connected compact oriented $d$-manifold (smooth), then $HH^{\ast}(C^{\ast}(M))$ the Hochschild ...
7
votes
0answers
129 views

Bang's open question strengthening Tarski's planks problem

Tarski's Planks problem, solved by Thøger Bang in 1951, says (in a simplified $\mathbb{R}^2$ version) that it requires "planks" (parallel strips) of total width $\ge d$ in order to completely cover a ...
4
votes
1answer
175 views

Sets not containing the vertices of unit triangles (Question posed by Erdős)

Following this post, I have been thinking about the problem posed by Erdős, Does there exist a constant $c > 0$ such that every subset $A$ of the plane of area more than $c$ contains the ...
5
votes
0answers
93 views

Functions on [0,1] with positive fractional series coefficients and symmetric under x->1-x

Suppose a function $g(x)$ has a convergent power series expansion with real (not necessarily integer or rational) exponents on $[0,1]$ of the form $$ g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i ...
6
votes
1answer
285 views

Conway's subprime Fibonacci sequences

I want to be certain I have the latest information on Conway's subprime Fibonacci sequences, arXiv-posted a year ago; I am referencing the status in a review. To wit, starting with $(0,1)$:1 $$ 0, 1, ...
20
votes
1answer
976 views

Why is it so hard to prove Toeplitz' conjecture?

I'm a layman in mathematics, so please excuse me in advance for anything in this question that may be inappropriate :D. Well: Four years ago, I was reading (and working to solve the puzzles on) ...
-6
votes
3answers
183 views

Collecting sufficient conditions for Sorli's conjecture on odd perfect numbers [closed]

Sorli's conjecture predicts that, for an odd perfect number $N$ given in the Eulerian form $N = {q^k}{n^2}$ (where $q$ is prime with $\gcd(q, n) = 1$ and $q \equiv k \equiv 1 \pmod 4$), the condition ...
3
votes
1answer
296 views

Open problems books [closed]

As the title might indicate , I would like to look for recommendations for mathematical book that present open problems in depth with commentary. The only book of this type that I've come across is ...
1
vote
0answers
246 views

On Descartes / spoof odd perfect numbers

Descartes, Frenicle, and subsequently Sorli, conjectured that $k = 1$, if $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and ...
2
votes
0answers
203 views

Are there any interesting open questions having to do with submodularity, specially in the intersection of theoretical machine learning? [closed]

I was interested in knowing about open research topics related with sub modularity, specially within its intersection with theoretical machine learning (and related topics). It seems to me that much ...
0
votes
1answer
205 views

Reference request: Research done on whether the Euler prime can be the largest factor of an odd perfect number

(Note: This was cross-posted from MSE.) I posted the following reference request in MSE three (3) days ago, but was unable to elicit any responses. I am cross-posting it to MO, hoping that it is ...
9
votes
2answers
482 views

Can any finite lattice be realized as an intermediate subgroups lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$. Question: Can any finite lattice be realized as ...
19
votes
5answers
2k views

Collection of conjectures and open problems in graph theory

Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?
5
votes
1answer
394 views

Ruth-Aaron triples, etc

A Ruth-Aaron pair is two numbers $(n,n+1)$ such that their sum of prime factors is equal, counting repeated prime factors. (The name refers to Hank Aaron's 715 homeruns surpassing Babe Ruth's 714!) So ...
13
votes
0answers
380 views

Is every projective space curve a set-theoretic intersection of two surfaces? What is known about this question?

I am sorry if this question has been already asked, couldn't find anything similar myself. I have recently recalled this long standing open problem of whether every irreducible curve in ...
2
votes
0answers
89 views

Long paths in the supercritical percolation.

I have a question on the length of the longest path, denoted by $\ell_n$, in the supercritical percolation on $[0,n]^d$, denoted by $C_n$. We know that $C_n$ has a giant component whose size is of ...
1
vote
2answers
471 views

Improving the bound $q < n\sqrt{3}$ for an odd perfect number $N = {q^k}{n^2}$ given in Eulerian form [closed]

(Note: This was cross-posted from MSE.) Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1$). Therefore, ...
4
votes
0answers
149 views

Reduction argument from a general vertex set V(G) to a prime power in Prof. Keevash's proof on the Existence of Designs

The proof flow of the paper "On the Existence of Designs" by Prof. Keevash as I understand it is the following: -- Reduction from the general case to $V = \mathbb{F}_{p^a}$ (Lemma 6.3) -- Covering ...
6
votes
1answer
227 views

Inverted pair of complex analytic families

I read the following "problem" in an old set of notes of Morrow and Kodaira which focused on deformations of complex manifolds: Find a pair of complex analytic families $\lbrace M_t\rbrace$ and ...
9
votes
0answers
183 views

Randomly placing nonoverlapping unit cuboids

Suppose one places unit cuboids of dimension $d$ with min-corners uniformly distributed to lie in $[0,n]^d$, but with cuboid (strict) overlap forbidden. At some point, the region is "saturated," ...
2
votes
2answers
331 views

Gauss Codes that produce classical knots as opposed to virtual knots

I have been doing some research in Gauss codes and have been reading Kauffman's paper Virtual Knot Theory. In section 3.3, Theorem 2, he states that If $K$ is a virtual knot whose underlying Gauss ...
14
votes
0answers
348 views

Does $S^4$ have a “symplecto-homeomorphic” structure?

The 4-sphere cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms ...
34
votes
3answers
1k views

Is the fixed point property for posets preserved by products?

Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point. Theorem. Suppose $P$ and $Q$ are posets ...
3
votes
1answer
131 views

reduction to np hard ordering problem

I am trying to show a reduction from a problem of ordering problem to an np-hard problem that has approximation poly-time algorithm. My problem is: I have M auctions and in each auction I have N ...
1
vote
1answer
116 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 ...
1
vote
0answers
167 views

Both NP-hard but different [closed]

What's the fundamental difference between the Knapsack problem and the travelling salesman (TSP) problem both of which are NP-hard, while the reality is that TSP could be solved much much faster?
7
votes
2answers
478 views

Update on list of open problems for Cherednik/Symplectic Reflection Algebras

Background: There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & ...
7
votes
4answers
1k views

Is the Manickam-Miklós-Singhi Conjecture solved? [closed]

This arXiv paper is claimed to contain a proof for the MMS conjecture. But it seems that this manuscript is not yet peer reviewed by other mathematicians. I personally tried to follow the paper, but ...
10
votes
1answer
312 views

Subspaces of $l_{1}$ are not Lipschitz complemented in $l_{1}$

I have thought about the following question for several years. This question may be stupid or not interesting. My question is: Is there a subspace $U$ of $l{_1}$ such that the quotient $l_{1}/U$ is ...