Number Theory

Special Primes ★

Author(s): George BALAN

Conjecture   Let $ p $ be a prime natural number. Find all primes $ q\equiv1\left(\mathrm{mod}\: p\right) $, such that $ 2^{\frac{\left(q-1\right)}{p}}\equiv1\left(\mathrm{mod}\: q\right) $.

Keywords:

Posted by maththebalans
updated February 7th, 2013
2 comments
Topology

Hilbert-Smith conjecture ★★

Author(s): David Hilbert; Paul A. Smith

Conjecture   Let $ G $ be a locally compact topological group. If $ G $ has a continuous faithful group action on an $ n $-manifold, then $ G $ is a Lie group.

Keywords:

Posted by porton
updated February 6th, 2011
add new comment
Topology

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
Topology

Sticky Cantor sets ★★

Author(s):

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