Re: Collatz at the Hilbert Hotel

From: Michael Collins (michael.collins_at_arm.com)
Date: 01/14/05 

Date: Fri, 14 Jan 2005 00:21:34 +0000

"mensanator@aol.compost" wrote:

> 01001011111010000000000000000000000000000000000000
> 01110001111000000000000000000000000000000000000000
> 10101010111000000000000000000000000000000000000000
> 00000000011000000000000000000000000000000000000001<--edge wrapping
> 10000000101000000000000000000000000000000000000001 shows how the
> 01000001000000000000000000000000000000000000000010 pattern is
> 01100010000000000000000000000000000000000000000011 independent of
> any mathematical anchor. this is simply a circular array.

I'd not thought about the wrapping of lines but surely once you've wrapped you're no
longer operating on the lowest significant bit.

The next statement's great:-

> As long as the array is larger than the Excursion, the CA will generate the same
> bit patterns as the standard Collatz algorithm

"As long as the array is larger than the Excursion"
This is a major part of the proof for the Collatz conjecture, if it's possible to
derive a maximum excursion from any initial point then you're showing that you can
never head off to infinity.

For any sequence in the standard collatz if you can show a minimum (that isn't 1)
then you have to have a closed loop or a sequence heading for infinity.

If you've got a maximum size for the excursion then it can't be the latter so you've
only got the loop to worry about - not that it's "only" of course :-).

Regards,
Mike... 

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