S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
It is currently Wed, 22 Oct 2014 04:08:53 UTC

All times are UTC [ DST ]




Post new topic Reply to topic  [ 3 posts ] 
Author Message
 Post subject: Modular Arithmetic (1)
PostPosted: Tue, 13 Oct 2009 19:41:55 UTC 
Offline
Moderator
User avatar

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 14139
Location: Austin, TX
(Taken from the Pi Mu Epsilon Journal's Fall 2009 Problem Set)

Show that \exists\alpha\in\mathbb{R}\;\forall n\in\mathbb{N}, \lfloor \alpha^n\rfloor\equiv n\pmod{5}

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination


Top
 Profile  
 
PostPosted: Sat, 24 Oct 2009 17:08:48 UTC 
Online
Moderator
User avatar

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6932
Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
Shadow wrote:
(Taken from the Pi Mu Epsilon Journal's Fall 2009 Problem Set)

Show that \exists\alpha\in\mathbb{R}\;\forall n\in\mathbb{N}, \lfloor \alpha^n\rfloor\equiv n\pmod{5}


There is nothing special about mod 5...

_________________
\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


Top
 Profile  
 
PostPosted: Sun, 25 Oct 2009 01:42:09 UTC 
Offline
Moderator
User avatar

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 14139
Location: Austin, TX
outermeasure wrote:
Shadow wrote:
(Taken from the Pi Mu Epsilon Journal's Fall 2009 Problem Set)

Show that \exists\alpha\in\mathbb{R}\;\forall n\in\mathbb{N}, \lfloor \alpha^n\rfloor\equiv n\pmod{5}


There is nothing special about mod 5...


shhhh.....

<_<
>_>

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination


Top
 Profile  
 
Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 3 posts ] 

All times are UTC [ DST ]


Who is online

Users browsing this forum: No registered users


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum

Search for:
Jump to:  
Contact Us | S.O.S. Mathematics Homepage
Privacy Statement | Search the "old" CyberBoard

users online during the last hour
Powered by phpBB © 2001, 2005-2011 phpBB Group.
Copyright © 1999-2013 MathMedics, LLC. All rights reserved.
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA