Open Problem Garden

Nonseparating planar continuum ★★

Author(s):

Conjecture Does any path-connected, compact set in the plane which does not separate the plane have the fixed point property?

A set has the fixed point property if every continuous map from it into itself has a fixed point.

Keywords: fixed point

Posted by porton
updated February 16th, 2011

1 comment

Conjecture Let $C$ be a Cantor set embedded in $\mathbb{R}^n$ . Is there a self-homeomorphism $f$ of $\mathbb{R}^n$ for every $\epsilon$ greater than $0$ so that $f$ moves every point by less than $\epsilon$ and $f(C)$ does not intersect $C$ ? Such an embedded Cantor set for which no such $f$ exists (for some $\epsilon$ ) is called "sticky". For what dimensions $n$ do sticky Cantor sets exist?

Keywords: Cantor set

Posted by porton
updated April 10th, 2012

2 comments

Achahboun Chakir

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Number Theory

Polignac's Conjecture ★★★

Author(s): de Polignac

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