**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

1k 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 ...

**19**

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

766 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

462 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 ...

**4**

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

504 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

820 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 ...

**5**

votes

**1**answer

829 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

662 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

753 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 ...

**30**

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

751 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

420 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

351 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 ...

**4**

votes

**0**answers

499 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 ...

**25**

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

446 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 ...

**21**

votes

**1**answer

1k 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 ...

**19**

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 $d$. ...

**4**

votes

**1**answer

424 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

713 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 ...

**8**

votes

**2**answers

531 views

### Base locus of divisors on blowings up of the projective space

Let $X$ be the blowing-up of $\mathbb{P}^n$ in $r$ points in general position.
Let $\{H,E_1,...,E_r\}$ be the standard basis of $Pic(X)$. Further let $D=dH-\sum m_j E_j$, be an effective divisor, with ...

**26**

votes

**0**answers

1k views

### A Combinatorial Abstraction for The “Polynomial Hirsch Conjecture”

Consider $t$ disjoint families of subsets of {1,2,â€¦,n}, ${\cal F}_1,{\cal F_2},\dots {\cal F_t}$ .
Suppose that
(*)
For every $i \lt j \lt k$
and every $R \in {\cal F}_i$, and $T \in {\cal F}_k$, ...

**22**

votes

**2**answers

2k views

### Projective Plane of Order 12

I asked this question on the new Theoretical Computer Science "overflow" site, and commenters suggested I ask it here. That question is here, and it contains additional links, which I doubt I can ...

**20**

votes

**6**answers

1k views

### What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $\Bbb Q$?

Some experts have a hunch that for any nonnegative integer $r$ there are infinitely many elliptic curves over $\Bbb Q$ with Mordell-Weil rank at least $r$.
The best empirical evidence for this hunch ...

**15**

votes

**2**answers

2k views

### Walking to infinity on the primes: The prime-spiral moat problem

It is an unsolved problem to decide if it is possible to "walk to infinity" from the origin
with bounded-length steps, each touching a Gaussian prime as a stepping stone.
The paper by Ellen Gethner, ...

**31**

votes

**2**answers

2k views

### Euler and the Four-Squares Theorem

There are several questions in the Euler-Goldbach correspondence that
I am unable to answer. Sometimes it does not take very much: in his
letter to Goldbach dated June 9th, 1750, Euler conjectured
...

**2**

votes

**1**answer

898 views

### The importance of Poincare Conjecture or SPC4?

As well known, Perelman proved Poincare conjecture by proving Thurston's Geometrization conjecture.
Somebody says that we can understand part of the universe from Poincare conjecture.
As a purely ...

**13**

votes

**2**answers

965 views

### Detecting almost-primes quickly

There are many fast algorithms (deterministic and probabilistic) for detecting primality. Are there any fast algorithms (probabilistic ones allowed) known for detecting whether a number is the product ...

**20**

votes

**0**answers

1k views

### Finite-dimensional subalgebras of $C^\star$-algebras

Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. ...

**11**

votes

**2**answers

2k views

### What impact would P!=NP have on the characterization of BQP?

Many complexity theorists assume that $P\ne NP.$ If this is proved, how would it impact quantum computing and quantum algorithms? Would the proof immediately disallow quantum algorithms from ever ...

**12**

votes

**0**answers

2k views

### How can an approach to $P$ vs $NP$ based on descriptive complexity avoid being a natural proof in the sense of Raborov-Rudich?

EDIT: This question has been modified to make it a stand-alone question. Feel free to retract your votes for the previous version.
Here are Vinay Deolalikar's paper, and Richard Lipton's first post ...

**4**

votes

**1**answer

595 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

**3**answers

1k views

### Conjugacy classes and reduced group $C^*$-algebra of an amenable group

The reduced $C^*$-algebra of a non-abelian free group $G$ has a unique trace. Hence, there is no chance to separate conjugacy classes of group elements using traces on $C^\star_{red} G$. On the other ...

**11**

votes

**3**answers

2k views

### Permute Wada Lakes keeping the coastline intact? (still open in dim >2)

Wada Lakes are three disjoint open subsets of $\mathbb R^2$ with common boundary. Originally they were constructed by hand, but they also arise naturally in the real life, that is, theory of dynamical ...

**16**

votes

**1**answer

1k views

### Are all primes in a PAP-3?

Van der Corput [1] proved that there are infinitely many arithmetic progressions of primes of length 3 (PAP-3). (Green & Tao [2] famously extended this theorem to length $k$.)
But taking this in ...

**27**

votes

**3**answers

3k views

### Can we cover the unit square by these rectangles?

The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1.
It is easy to show that
$$\sum_{1 \leq k } ...

**0**

votes

**1**answer

1k views

### Existence of a pseudo-polynomial time algorithm for a counting problem.

Let T={1,...,n} be a set of tasks. Each task i has associated a non negative processing time p_i and a deadline d_i. A feasible schedule of the tasks consists of a permutation of n elements pi, such ...

**12**

votes

**0**answers

1k views

### Razborov's response to Almost Natural Proofs

This post is about Natural Proofs barrier in computational complexity.
There are two recent papers related to this. They are:
Amplifying lower bounds by means of self-reducibility by Eric Allender ...

**3**

votes

**1**answer

704 views

### Cyl(E) = Borel(E) for E non-reflexive Grothendieck Banach space

This is sort of a follow-up to Borel(X) = \sigma(X') for X non-separable
PROBLEM: Given a Banach space $E$ over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$ that has the Grothendieck property. ...

**9**

votes

**6**answers

1k views

### Smallest area shape that covers all unit length curve

On a euclidean plane, what is the minimal area shape S, such that for every unit length curve, a translation and a rotation of S can cover the curve.
What are the bounds of the shape's area if this ...

**16**

votes

**6**answers

2k views

### Question on consecutive integers with similar prime factorizations

Suppose that $n=\prod_{i=1}^{k} p_i^{e_i}$ and $m=\prod_{i=1}^{l} q_i^{f_i}$ are prime factorizations of two positive integers $n$ and $m$, with the primes permuted so that $e_1 \le e_2 \cdots \le ...

**37**

votes

**13**answers

5k views

### What are some of the big open problems in 3-manifold theory?

From what I understand, the geometrization theorem and its proof helped to settle a lot of outstanding questions about the geometry and topology of 3-manifolds, but there still seems to be quite a lot ...

**5**

votes

**2**answers

560 views

### Lower bounds (or less) for the period of \sqrt(D) and related sequences.

This is a continuation of
Lower bounds for period length of continued fraction of square root
which is a continuation of
Upper bound of period length of continued fraction representation of very ...

**20**

votes

**2**answers

2k views

### Curves of constant curvature on S^2

Most probably this is a well known question.
Consider $S^2$ with a Riemannian metric. I would like to ask what is known about the structure of the set of simple (without self-intersections) closed ...

**47**

votes

**8**answers

6k views

### The “sensitivity” of 2-colorings of the d-dimensional integer lattice

Consider the $d$-dimensional integer lattice, $Z^d$. Call two points in $Z^d$ "neighbors" if their Euclidean distance is 1 (i.e., if they differ by 1 on exactly one coordinate).
Let $C$ be a ...

**11**

votes

**2**answers

2k views

### Does listing the prime factors always stop?

Take a natural number's prime factors and list them increasingly and repeating them according to multiplicity. Concatenate their decimal (or in any base) representation to get a new number and repeat ...

**23**

votes

**4**answers

1k views

### Is Lebesgue's “universal covering” problem still open?

The following problem has been attributed to Lebesgue. Let "set" denote any subset of the Euclidean plane. What is the greatest lower bound of the diameter of any set which contains a subset congruent ...