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Number Theory » Analytic N.T.

Are there infinite number of Mersenne Primes? ★★★★

Author(s):

Conjecture A Mersenne prime is a Mersenne number $M_n = 2^p - 1$ that is prime.

Are there infinite number of Mersenne Primes?

Keywords: Mersenne number; Mersenne prime

Posted by kurtulmehtap
updated February 4th, 2015

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Number Theory » Analytic N.T.

Are all Mersenne Numbers with prime exponent square-free? ★★★

Author(s):

Conjecture Are all Mersenne Numbers with prime exponent ${2^p-1}$ Square free?

Keywords: Mersenne number

Posted by kurtulmehtap
updated February 4th, 2015

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Topology

What are hyperfuncoids isomorphic to? ★★

Author(s): Porton

Let $\mathfrak{A}$ be an indexed family of sets.

Products are $\prod A$ for $A \in \prod \mathfrak{A}$ .

Hyperfuncoids are filters $\mathfrak{F} \Gamma$ on the lattice $\Gamma$ of all finite unions of products.

Problem Is $\bigcap^{\mathsf{\tmop{FCD}}}$ a bijection from hyperfuncoids $\mathfrak{F} \Gamma$ to:

$\mathfrak{A}$

If yes, is $\operatorname{up}^{\Gamma}$ defining the inverse bijection? If not, characterize the image of the function $\bigcap^{\mathsf{\tmop{FCD}}}$ defined on $\mathfrak{F} \Gamma$ .

Consider also the variant of this problem with the set $\Gamma$ replaced with the set $\Gamma^{\ast}$ of complements of elements of the set $\Gamma$ .

Keywords: hyperfuncoids; multidimensional

Posted by porton
updated December 9th, 2014

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Topology

Another conjecture about reloids and funcoids ★★

Author(s): Porton

Definition $\square f = \bigcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f$ for reloid $f$ .

Conjecture $(\mathsf{RLD})_{\Gamma} f = \square (\mathsf{RLD})_{\mathrm{in}} f$ for every funcoid $f$ .

Note: it is known that $(\mathsf{RLD})_{\Gamma} f \ne \square (\mathsf{RLD})_{\mathrm{out}} f$ (see below mentioned online article).

Keywords:

Posted by porton
updated November 28th, 2014

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Analysis

Inequality for square summable complex series ★★

Author(s): Retkes

Conjecture For all $\alpha=(\alpha_1,\alpha_2,\ldots)\in l_2(\cal{C})$ the following inequality holds $\sum_{n\geq 1}|\alpha_n|^2\geq \frac{6}{\pi^2}\sum_{k\geq0}\bigg| \sum_{l\geq0}\frac{1}{l+1}\alpha_{2^k(2l+1)}\bigg|^2$

Keywords: Inequality

Posted by tigris35711
updated October 29th, 2014

2 comments

Theoretical Comp. Sci. » Complexity

One-way functions exist ★★★★

Author(s):

Conjecture One-way functions exist.

Keywords: one way function

Posted by porton
updated October 12th, 2014

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Graph Theory

Chromatic Number of Common Graphs ★★

Author(s): Hatami; Hladký; Kráľ; Norine; Razborov

Question Do common graphs have bounded chromatic number?

Keywords: common graph

Posted by David Wood
updated August 15th, 2014

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Number Theory

Erdős–Straus conjecture ★★

Author(s): Erdos; Straus

Conjecture

For all $n > 2$ , there exist positive integers $x$ , $y$ , $z$ such that $1/x + 1/y + 1/z = 4/n$ .

Keywords: Egyptian fraction

Posted by ACW
updated July 14th, 2014

4 comments

Number Theory » Combinatorial N.T.

The 3n+1 conjecture ★★★

Author(s): Collatz

Conjecture Let $f(n) = 3n+1$ if $n$ is odd and $\frac{n}{2}$ if $n$ is even. Let $f(1) = 1$ . Assume we start with some number $n$ and repeatedly take the $f$ of the current number. Prove that no matter what the initial number is we eventually reach $1$ .

Keywords: integer sequence

Posted by dododododo
updated July 11th, 2014

4 comments

Graph Theory » Coloring » Vertex coloring

List Hadwiger Conjecture ★★

Author(s): Kawarabayashi; Mohar

Conjecture Every $K_t$ -minor-free graph is $c t$ -list-colourable for some constant $c\geq1$ .

Keywords: Hadwiger conjecture; list colouring; minors

Posted by David Wood
updated July 7th, 2014

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Number Theory

Lucas Numbers Modulo m ★★

Author(s):

Conjecture The sequence {L(n) mod m}, where L(n) are the Lucas numbers, contains a complete residue system modulo m if and only if m is one of the following: 2, 4, 6, 7, 14, 3^k, k >=1.

Keywords: Lucas numbers

Posted by Martin Erickson
updated July 1st, 2014

1 comment

Number Theory » Combinatorial N.T.

Divisibility of central binomial coefficients ★★

Author(s): Graham

Problem (1) Prove that there exist infinitely many positive integers $n$ such that $\gcd({2n\choose n}, 3\cdot 5\cdot 7) = 1.$

Problem (2) Prove that there exists only a finite number of positive integers $n$ such that $\gcd({2n\choose n}, 3\cdot 5\cdot 7\cdot 11) = 1.$

Keywords:

Posted by maxal
updated June 29th, 2014

5 comments

Graph Theory » Hypergraphs

¿Are critical k-forests tight? ★★

Author(s): Strausz

Conjecture

Let $H$ be a $k$ -uniform hypergraph. If $H$ is a critical $k$ -forest, then it is a $k$ -tree.

Keywords: heterochromatic number

Posted by Dino
updated May 5th, 2014

1 comment

Combinatorics » Posets

Saturated $k$-Sperner Systems of Minimum Size ★★

Author(s): Morrison; Noel; Scott

Question Does there exist a constant $c>1/2$ and a function $n_0(k)$ such that if $|X|\geq n_0(k)$ , then every saturated $k$ -Sperner system $\mathcal{F}\subseteq \mathcal{P}(X)$ has cardinality at least $2^{(1+o(1))ck}$ ?

Keywords: antichain; extremal combinatorics; minimum saturation; saturation; Sperner system

Posted by Jon Noel
updated April 12th, 2014

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Graph Theory » Coloring » Vertex coloring

List Colourings of Complete Multipartite Graphs with 2 Big Parts ★★

Author(s): Allagan

Question Given $a,b\geq2$ , what is the smallest integer $t\geq0$ such that $\chi_\ell(K_{a,b}+K_t)= \chi(K_{a,b}+K_t)$ ?

Keywords: complete bipartite graph; complete multipartite graph; list coloring

Posted by Jon Noel
updated April 12th, 2014

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Geometry

Generalised Empty Hexagon Conjecture ★★

Author(s): Wood

Conjecture For each $\ell\geq3$ there is an integer $f(\ell)$ such that every set of at least $f(\ell)$ points in the plane contains $\ell$ collinear points or an empty hexagon.

Keywords: empty hexagon

Posted by David Wood
updated March 26th, 2014

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Geometry

General position subsets ★★

Author(s): Gowers

Question What is the least integer $f(n)$ such that every set of at least $f(n)$ points in the plane contains $n$ collinear points or a subset of $n$ points in general position (no three collinear)?

Keywords: general position subset, no-three-in-line problem

Posted by David Wood
updated March 24th, 2014

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Graph Theory » Basic G.T. » Minors

Forcing a 2-regular minor ★★

Author(s): Reed; Wood

Conjecture Every graph with average degree at least $\frac{4}{3}t-2$ contains every 2-regular graph on $t$ vertices as a minor.

Keywords: minors

Posted by David Wood
updated March 16th, 2014

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Graph Theory

Fractional Hadwiger ★★

Author(s): Harvey; Reed; Seymour; Wood

Conjecture For every graph $G$ ,
(a) $\chi_f(G)\leq\text{had}(G)$
(b) $\chi(G)\leq\text{had}_f(G)$
(c) $\chi_f(G)\leq\text{had}_f(G)$ .

Keywords: fractional coloring, minors

Posted by David Wood
updated March 16th, 2014

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Topology

Generalized path-connectedness in proximity spaces ★★

Author(s): Porton

Let $\delta$ be a proximity.

A set $A$ is connected regarding $\delta$ iff $\forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \} : \left( X \cup Y = A \Rightarrow X \mathrel{\delta} Y \right)$ .