Open Problem Garden

Polignac's Conjecture ★★★

Conjecture Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.

In particular, this implies:

Conjecture Twin Prime Conjecture: There are an infinite number of twin primes.

Keywords: prime; prime gap

Posted by Hugh Barker
updated June 18th, 2013

3 comments

combinatorial geometry, visibility graph

Geometry

Point sets with no empty pentagon ★

Author(s): Wood

Problem Classify the point sets with no empty pentagon.

Keywords: combinatorial geometry; visibility graph

Posted by David Wood
updated March 25th, 2019

2 comments

finite algebra

congruence lattice

Algebra

Finite Lattice Representation Problem ★★★★

Author(s):

Conjecture

There exists a finite lattice which is not the congruence lattice of a finite algebra.

Keywords: congruence lattice; finite algebra

Posted by williamdemeo
updated December 10th, 2010

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Topology

Outer reloid of restricted funcoid ★★

Author(s): Porton

Question $( \mathsf{RLD})_{\mathrm{out}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{out}} f) \cap^{\mathsf{RLD}} ( \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B})$ for every filter objects $\mathcal{A}$ and $\mathcal{B}$ and a funcoid $f\in\mathsf{FCD}(\mathrm{Src}\,f; \mathrm{Dst}\,f)$ ?

Keywords: direct product of filters; outer reloid

Posted by porton
updated December 3rd, 2010

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image

domain

Topology

Domain and image of inner reloid ★★

Author(s): Porton

Conjecture $\ensuremath{\operatorname{dom}}( \mathsf{\ensuremath{\operatorname{RLD}}})_{\ensuremath{\operatorname{in}}} f =\ensuremath{\operatorname{dom}}f$ and $\ensuremath{\operatorname{im}}( \mathsf{\ensuremath{\operatorname{RLD}}})_{\ensuremath{\operatorname{in}}} f =\ensuremath{\operatorname{im}}f$ for every funcoid $f$ .