**34**

votes

**4**answers

2k views

### The maximum of a polynomial on the unit circle

Encouraged by the progress made in a recently posted MO problem, here is a "conceptually related" problem originating from a 2003 joint paper of Sergei Konyagin and myself.
Suppose we are given $n$ ...

**11**

votes

**3**answers

2k views

### Zero divisor conjecture for finite fields

I believe that the most attractive "zero-divisor" conjecture is the existence of non-trivial zero-divisors in a group ring $\mathbb{C}[G]$ for a torsion free group $G$. For the sake of knowledge let ...

**13**

votes

**3**answers

1k views

### Meromorphic 1-form and Picard's theorem

Let $D$ be the open unit disk in the complex plane and $U_1,U_2,\,\ldots\,,U_n$ be an open cover of the puntured disk $D^*= D\setminus\{0\}$. Suppose on each open set $U_j$ there is an injective ...

**3**

votes

**0**answers

558 views

### Fixed point theorem for convex, closed multivalued mapping

There is well-known fixed point theorem theorem for multivalued l.s.c. maps, based on Michael selection theorem:
Suppose, that $X$ is compact, convex and metrizable in locally convex Hausdorff ...

**7**

votes

**1**answer

413 views

### Symplectic structures on a homotopy complex projective space

For $n>2$, there are infinitely many differentiable structures on the homotopy type of $\mathbb{C}P^n$. I want to know which differentiable structures support a symplectic form.
My question is as ...

**4**

votes

**1**answer

830 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 ...

**24**

votes

**4**answers

1k views

### Trees in groups of exponential growth

Question: Let $G$ be a finitely generated group with exponential growth.
Is there a finite generating set $S \subset G$, such that the associated Cayley graph $Cay(G,S)$ contains a binary tree?
...

**4**

votes

**1**answer

355 views

### A question concerning products of finite cyclic groups

Let $m_1,\ldots, m_n$ be pairwise coprime natural numbers $\geq 1$. We consider the product $$G(m_1,\ldots,m_n) := \prod_{i=1}^{n} \mathbb{Z} / m_i \mathbb{Z}.$$ We define $M(n)$ as the set $n$-tuple ...

**23**

votes

**1**answer

1k views

### Can you flip the end of a large exotic $\mathbb{R}^4$

Can you flip the end of a large exotic $\mathbb{R}^4$
Background
Definition (Exotic $\mathbb{R}^4$):
An exotic $\mathbb{R}^4$ is a smooth manifold $R$ homeomorphic but not diffeomorphic to ...

**3**

votes

**3**answers

759 views

### Bounds on number of conjugacy classes in terms of number of elements of a group ?

What are bounds on number of conjugacy classes in terms of number of elements of a group ?
(I allowed myself to edit the question in spirit of remarkable answers given to it by Gerry Myerson and ...

**24**

votes

**4**answers

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 ...

**7**

votes

**3**answers

1k views

### Kronecker Approximation theorem and Fibonacci numbers

There is a famous old theorem by Kronecker that for every positive real $\alpha$ and $\epsilon>0$ there exists a positive integer n such that $\alpha n$ is within $\epsilon$ of an integer.
...

**0**

votes

**2**answers

525 views

### A simple question regarding the sum-of-divisors function

A good day to everyone.
Consider the following "Conjecture":
If $a, b \in M \subset \mathbb{N}$, then $1 < a < b$ and ... [plus some more conditions on $a, b$ and $M$...] if and only if ...

**0**

votes

**1**answer

371 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

**1**answer

301 views

### A question on a special type of function

Suppose I have a function $f$, and positive integers $x$ and $y$ such that $x$ is a
square, $x \ne y$ and $y \ne \sqrt{x}$.
Now, assume that:
$|\frac{f(x)}{y} - \frac{f(y)}{x}| > 2$
...

**13**

votes

**3**answers

962 views

### Card game / options pricing / Brownian bridge question

We play a game. I shuffle a deck of cards and start dealing them face up. After any card you can say "stop", at which point I pay you 1 dollar for every red card dealt and you pay me 1 for every black ...

**17**

votes

**1**answer

830 views

### What's the Kirby Diagram of a universal $\mathbb{R}^4$?

What's the Kirby diagram of a universal $\mathbb{R}^4$?
Background
Define $\mathcal{R}$ as the set of smoothings of $\mathbb{R}^4$. For two oriented elements $R_1$, $R_2$ in $\mathcal{R}$ we can ...

**13**

votes

**0**answers

466 views

### Comparing the Kazhdan-Lusztig and Steinberg pre-orders

Both Kazhdan-Lusztig and Steinberg have defined pairs of preorders on $S_n$. Kazhdan and Lusztig's preorders come from their basis:
We write $x\leq_L y$ if any left ideal spanned by K-L basis ...

**0**

votes

**1**answer

348 views

### Something new in old question about sums of three polynomial cubes ?

An old problem asks whether or not the polynomial
$$
t \in \mathbb{Q}[t]
$$
is a sum of three cubes, (of polynomials in $\mathbb{Q}[t]$).
Question: Something new known now ?
Somebody has an idea of ...

**14**

votes

**2**answers

742 views

### Free subgroups vs law

Consider the following two conditions for a group $G$:
(1) $G$ does not satisfy a nontrivial law.
(2) $G$ contains a non-abelian free subgroup.
Obviously (2) implies (1) and it is easy to ...

**56**

votes

**1**answer

3k views

### Nontrivial finite group with trivial group homologies?

I stumbled across this question in a seminar-paper a long time ago:
Does there exist a positive integer $N$ such that if $G$ is a finite group with $\bigoplus_{i=1}^NH_i(G)=0$ then $G=\lbrace ...

**14**

votes

**1**answer

602 views

### Amenability of groups in terms of a perturbation condition

Let $G$ be a countable group and $\lambda \colon G \to U(\ell^2 G)$ its left-regular representation. Suppose that there exists a constant $C>0$ such that for all $T \in B(\ell^2 G)$
$$\inf ...

**4**

votes

**1**answer

426 views

### Implication of Polignac's conjecture on prime distribution in models of PA

Polignac's conjecture (PC) is that there exists infinitely many pairs of consecutive prime numbers that are a distance $d$ apart for some natural number $d$. The twin prime conjecture is the ...

**15**

votes

**9**answers

4k views

### Open Questions in Riemannian Geometry

What are some major open problems in Riemannian Geometry? I tried googling it, but couldn't find any resources.

**19**

votes

**2**answers

1k views

### unboundedness of number of integral points on elliptic curves?

If $E/\mathbf{Q}$ is an elliptic curve and we put it into minimal Weierstrass form, we can count how many integral points it has. A theorem of Siegel tells us that this number $n(E)$ is finite, and ...

**6**

votes

**0**answers

260 views

### Real approximation for homogeneous spaces of linear algebraic groups

Let $X$ be a smooth geometrically integral variety over $\mathbf{Q}$, having a $\mathbf{Q}$-point.
We say that $X$ has the real approximation property if $X(\mathbf{Q})$ is dense in $X(\mathbf{R})$.
...

**8**

votes

**3**answers

1k views

### Prepending strings to primes.

Hello,
we all know that 31,331,3331,33331,333331,3333331,33333331 all are primes, and that 333333331 is not. Here we prepend the digit 3 to 31, to get a list of 7 primes.This gives me the following ...

**8**

votes

**1**answer

2k views

### The Invariant Subspace Problem: examples

Question. Is there a concrete example of a bounded linear operator on a Hilbert space for which it is not known if it has a non-trivial closed invariant subspace?
[Added 24.01.2011: According to ...

**20**

votes

**1**answer

2k views

### Iwasawa main conjectures vs Bloch-Kato conjectures

Let $p$ be a prime, $K$ be a number field, $S$ a finite set of finite places of $K$ containing the set $S_p$ of places above $p$ and the places at infinity, $G:=G_{K,S}$ the Galois group of the ...

**0**

votes

**1**answer

791 views

### Perfect Numbers - On Mersenne and Euler Primes

Hi, I apologize if there is already an (obvious) answer to my question, but please bear with me for the moment as I find it hard to see a good way to answer this question:
In the same way that the ...

**18**

votes

**2**answers

4k views

### Is the Invariant Subspace Problem interesting?

There's an amusing comment in Peter Lax's Functional Analysis book. After a brief description of the Invariant Subspace Problem, he says (paraphrasing) "...this question is still open. It is also an ...

**1**

vote

**1**answer

473 views

### Infinitely many pairs of primes?

Hi. I want to know how many (infinitely many) pairs of primes are known.
For convinience, let me give two definitions.
For any nonconstant polynomial $f(x)\in \mathbb{Z}[x]$, define $A_{f}=\lbrace ...

**3**

votes

**2**answers

2k views

### Re: Mordell's Equation $y^2 = x^3 + k$ and Perfect Numbers

I have already tried a somewhat exhaustive search of the literature, but couldn't find anything close to the problem that I am working on.
My question is: When does Mordell's Equation
$$y^2 = x^3 ...

**2**

votes

**1**answer

523 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 ...

**5**

votes

**2**answers

844 views

### On Sorli's Conjecture Re: OPNs (Circa 2003)

In the PhD dissertation titled "Algorithms in the Study of Multiperfect and Odd Perfect Numbers" (hyperlinked here) and completed in 2003, Ronald Sorli conjectured that the exponent $k$ on the Euler ...

**6**

votes

**1**answer

899 views

### A Game of Knights and Queens

Let $m,n,u,v \in \mathbb{N}$ be parameters with $m,n \geq 3$. Suppose two players play a game on a $m \times n$ chess board and we denote the squares of the board by the set of points $ (i,j) $ such ...

**4**

votes

**2**answers

675 views

### Artin's conjecture for n=2

I am interested in the following question:
Is it known that $2$ is a primitive root modulo $p$ for infinitely many primes $p$?
there is some information about Artin's conjecture in
...

**14**

votes

**1**answer

834 views

### Nonnegative to Positive Curvature.

This questions asks for your intuition and insight as I'm surprised by how little is known about the difference between nonnegative and positive curvature. I don't want to be completely vague, so I ...

**31**

votes

**2**answers

2k views

### Difficult examples for Frankl's union-closed conjecture

Frankl's well-known union-closed conjecture states that if F is a finite family of sets that is closed under taking unions (that is, if A and B belong to the family then so does $A\cup B$), then there ...

**16**

votes

**1**answer

814 views

### Covering the primes by arithmetic progressions

Define the length of a set of arithmetic progressions
of natural numbers
$A=\lbrace A_1, A_2, \ldots \rbrace$
to be $\min_i | A_i |$: the length of the shortest sequence
among all the progressions.
...

**1**

vote

**0**answers

426 views

### Zeroes of a tricky function.

I am attempting to show that there does not exist an N past which every open unit interval (k, k+1) -where k is an integer- contains a zero of the following function:
$h(x)=\sum_{n=2}^{[\sqrt(x)]} ...

**2**

votes

**0**answers

354 views

### Natural numbers n which satisfy gnu(n)=n?

Are there any natural numbers $n$ (other than 1) for which $gnu(n)=n$? We define $gnu(n)$ to
be the number of isomorphism classes of groups of order $n$. This question popped into my head today, and ...

**5**

votes

**0**answers

513 views

### Are there an infinite number of prime Euclid numbers?

A number defined as the product of first $n$ prime numbers $+1$ is called $n$th Euclid number. Are there any survey on the progress for answering the following question: are there an infinite number ...

**29**

votes

**1**answer

1k views

### Density of values of polynomials in two variables

This question is a reposting of a comment I made on Polynomial representing all nonnegative integers. Let $f(x,y)\in \mathbb Q[x,y]$ such that $f(\mathbb Z\times \mathbb Z)$ is a subset of $\mathbb ...

**8**

votes

**0**answers

492 views

### What polynomials biject from $\mathbb{N}^{2}$ to $\mathbb{N}$?

Perhaps there are none with integral coefficients; so let us admit rational coefficients. The map $(x, y) \mapsto x + \frac{1}{2}(x + y)(x + y + 1)$ is well known, and swapping $x$ and $y$ in the ...

**22**

votes

**1**answer

2k views

### Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle

Hillel Furstenberg conjectured that the only $2$-$3$-invariant probability measure on the circle without atoms is the Lebesgue measure. More precisely:
Question: (Furstenberg) Let $\mu$ be a ...

**20**

votes

**2**answers

2k views

### Is there a 7-regular graph on 50 vertices with girth 5? What about 57-regular on 3250 vertices?

The following problem is homework of a sort -- but homework I can't do!
The following problem is in Problem 1.F in Van Lint and Wilson:
Let $G$ be a graph where every vertex
has degree $d$. ...

**13**

votes

**1**answer

957 views

### The determinant of the sum of normal matrices

Given two normal matrices $A,B\in M_n({\mathbb C})$
whose respective spectra are $(\alpha_{1},\ldots,\alpha_{n})$ and
$(\beta_{1},\ldots,\beta_{n})$, is it true that $\det(A+B)$ belongs to
the ...

**4**

votes

**1**answer

436 views

### Milnor's isotopy invariant using spectral sequence?

I'm reading stalling's article "the augmented ideal in group ring" in Ann. Math. Studies 84(R. H. Fox memorial volume)
In his final remark, he says that Milnor's link invariant could be interpreted ...

**13**

votes

**0**answers

771 views

### Special values of Artin L-functions

This question might be naive and might carry the heuristic that we are living in the best possible world a little too far. If so, I appreciate being told so.
Background: Stark's conjecture interprets ...