# S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Sat, 25 May 2013 16:55:24 UTC

 All times are UTC [ DST ]

 Page 1 of 1 [ 12 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Awesome ProblemPosted: Fri, 11 May 2012 20:28:21 UTC
 Senior Member

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
Let and be positive integers. is defined inductively in the base -recursive. We first write in base e.g., as a sum of terms of the form , with . For each exponent , we write in the base -recursive, until all the numbers in the representation are less than . For example,

Let arbitrary. We define recursively, as following: if , we write in the base -recursive and we replace all the numbers for (even the exponents!), so we obtain the successor of . If , then . For example:

.
.
.

We say that the sequence "terminates" for some positive integer , if there exists a positive integer , such that when . Find all values of such that the sequence terminates when .

Hint:
Spoiler:
This isn't a very "ordinary" solution.

_________________
math puns are the first sine of madness
-JDR

Top

 Post subject: Re: Awesome ProblemPosted: Fri, 11 May 2012 20:54:45 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sun, 24 Jul 2005 20:12:39 UTC
Posts: 3692
Location: Ottawa Ontario
It gave me an awesome headache

_________________
I'm not prejudiced...I hate everybody equally!

Top

 Post subject: Re: Awesome ProblemPosted: Fri, 11 May 2012 23:23:19 UTC
 Senior Member

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
Denis wrote:
It gave me an awesome headache

lolol this problem tends to do that

_________________
math puns are the first sine of madness
-JDR

Top

 Post subject: Re: Awesome ProblemPosted: Fri, 11 May 2012 23:30:00 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12103
Location: Austin, TX
Denis wrote:
It gave me an awesome headache

Oh come now, Denis, it's not that bad. Are you sure you're thinking about it properly?

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

Top

 Post subject: Re: Awesome ProblemPosted: Sat, 12 May 2012 00:38:33 UTC
 Senior Member

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
Denis wrote:
It gave me an awesome headache

Oh come now, Denis, it's not that bad. Are you sure you're thinking about it properly?

Finding all of the numbers (with proof) definitely wasn't easy for me.

¿...Darn am I missing something, because my solution is ridiculous...?

EDIT: or are you trolling?~ I'm not very good at knowing who's a troll and who isn't over the internet. :/

_________________
math puns are the first sine of madness
-JDR

Top

 Post subject: Re: Awesome ProblemPosted: Sat, 12 May 2012 04:59:51 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6009
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Denis wrote:
It gave me an awesome headache

Oh come now, Denis, it's not that bad. Are you sure you're thinking about it properly?

Well, since Goodstein's theorem cannot be proven inside PA, unless one knows about ordinals and Cantor normal form etc. it is bound to be headaches. Indeed, if we believe PA is consistent, then PA+not(Goodstein) is also consistent and hence has a model, i.e. there exists nonstandard model of PA where Goodstein is false.

Anyway, I think OP meant , not .

_________________

Top

 Post subject: Re: Awesome ProblemPosted: Sat, 12 May 2012 07:36:56 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12103
Location: Austin, TX
outermeasure wrote:
Denis wrote:
It gave me an awesome headache

Oh come now, Denis, it's not that bad. Are you sure you're thinking about it properly?

Well, since Goodstein's theorem cannot be proven inside PA, unless one knows about ordinals and Cantor normal form etc. it is bound to be headaches.

Anyway, I think OP meant , not .

Perhaps I'm just used to slogging through things. And yes, is certainly what we want.

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

Top

 Post subject: Re: Awesome ProblemPosted: Sat, 12 May 2012 18:40:17 UTC
 Senior Member

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
outermeasure wrote:
Denis wrote:
It gave me an awesome headache

Oh come now, Denis, it's not that bad. Are you sure you're thinking about it properly?

Well, since Goodstein's theorem cannot be proven inside PA, unless one knows about ordinals and Cantor normal form etc. it is bound to be headaches. Indeed, if we believe PA is consistent, then PA+not(Goodstein) is also consistent and hence has a model, i.e. there exists nonstandard model of PA where Goodstein is false.

Anyway, I think OP meant , not .

Hmm PA is not used and x_1 is natural (sorry for the lack of tex I'm on my phone), but I'm not sure if you're seeing the solution? I don't know quite a bit of what you posted. My hint should have been
Spoiler:
My solution is not ordinary all in all.

_________________
math puns are the first sine of madness
-JDR

Top

 Post subject: Re: Awesome ProblemPosted: Sat, 12 May 2012 19:57:08 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12103
Location: Austin, TX
Oh really, Peano Arithmetic isn't used in a problem involving Peano Arithmetic? That would be impressive.

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

Top

 Post subject: Re: Awesome ProblemPosted: Sat, 12 May 2012 22:59:56 UTC
 Senior Member

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
rdj5933mile5math64 wrote:
but I'm not sure if you're seeing the solution?

Never mind, I looked up Goodstein's Theorem. Darn. A friend gave me this problem and solution, and I thought it would be nice to share. I had no idea it was a theorem.

_________________
math puns are the first sine of madness
-JDR

Top

 Post subject: Re: Awesome ProblemPosted: Sat, 12 May 2012 23:15:21 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12103
Location: Austin, TX
rdj5933mile5math64 wrote:
rdj5933mile5math64 wrote:
but I'm not sure if you're seeing the solution?

Never mind, I looked up Goodstein's Theorem. Darn. A friend gave me this problem and solution, and I thought it would be nice to share. I had no idea it was a theorem.

Most things you've seen are probably consequences of well-known theorems, don't be bummed, it's just you don't have the experience to see it.

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

Top

 Post subject: Re: Awesome ProblemPosted: Mon, 14 May 2012 15:05:39 UTC
 Senior Member

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
rdj5933mile5math64 wrote:
rdj5933mile5math64 wrote:
but I'm not sure if you're seeing the solution?

Never mind, I looked up Goodstein's Theorem. Darn. A friend gave me this problem and solution, and I thought it would be nice to share. I had no idea it was a theorem.

Most things you've seen are probably consequences of well-known theorems, don't be bummed, it's just you don't have the experience to see it.

lol thanks, I'm sorry that the problem I gave could be bazooka'd by a theorem.

_________________
math puns are the first sine of madness
-JDR

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 12 posts ]

 All times are UTC [ DST ]

#### Who is online

Users browsing this forum: No registered users

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

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous