Math Is Fun Forum

soroban

Member

Registered: 2007-03-09

Posts: 452

Yet Another Number Trap

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mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Yet Another Number Trap

[Jane had an attempted proof above this post]

Isn't the existence of other cycles a fairly famous open problem?
I'd be interested to see this proof you mention, soroban.

Jane, I'm not sure what you're doing. I can't see any mistakes in what you've written, but I also can't see you concluding anything. Sorry if I'm being dense.

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Why did the vector cross the road?
It wanted to be normal.

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Yet Another Number Trap

I was under the impression that this was unsolved:

http://en.wikipedia.org/wiki/Collatz_conjecture

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Yet Another Number Trap

Last result I know for it is (Oliveira e Silva 2008)

proved by computer for all numbers <= 5.47637715 × 10^18

Last edited by bobbym (2009-06-27 08:56:33)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Yet Another Number Trap

If M is odd, then we repeat from step one.

Let {x_n} be the sequence generated from your starting odd number.  So x_1 = 3*x_0 + 1 and x_2 = (3*x_0 + 1)/2.  Now in the case that x_2 is odd, we then have x_3 = (9*x_0 + 3)/2.  But the thing here is that x_3 > x_0.  If your M always turns out to be odd, then the sequence diverges to infinity.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Yet Another Number Trap

Actually, I’ve only proved that any number you start with will eventually be reduced to one that is congruent to 1 mod 3. I haven’t proved that it will eventually become 1.

But there is an upside to this: I feel confident in awarding you the longest proof ever of that fact for this problem.

3n+1 = 1 (mod 3)

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Yet Another Number Trap

I was referring to an ODD number ≡ 1 mod 3!

Jane, please don't delete your posts after they have been replied to.  It makes other people who are trying to follow this thread rather confused.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Yet Another Number Trap

Then please stop making pointless replies to my posts. I had already admitted that my hastily thought-up proof was neither here nor there, so there was no point at all in commenting on it.

Last edited by JaneFairfax (2009-06-27 22:56:27)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Yet Another Number Trap

Jane, mathsyperson posted:

Jane, I'm not sure what you're doing. I can't see any mistakes in what you've written, but I also can't see you concluding anything. Sorry if I'm being dense.

I was explaining to him what was wrong.  Don't be so embarrassed at making little mistakes.  I used the quadratic equation wrong in my algebra final not more than 2 months ago.  It happens.

At times you can learn more from mistakes.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Yet Another Number Trap

Such a mature and caring answer. Someone has a lot friends on this forum, hope that person comes to realize that.

Awful to hear about that error on a final. Everyone can relate. Only good that can come from it is you probably won't ever forget it.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Yet Another Number Trap

Okay, thanks, Ricky. You are nice. You do not insult me for making a typo, unlike integer.

What is wrong with integer anyway? When I made a careless typo  in another thread, he or she had to insult me by posting BS about my breakfast. There was absolutely no need for that!

Last edited by JaneFairfax (2009-07-26 01:07:19)

quittyqat

Member

Registered: 2009-04-08

Posts: 1,215

Re: Yet Another Number Trap

On most forums, what you just posted would not be allowed.

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I'll be here at least once every decade.