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Topic: This Week's Finds in Mathematical Physics (Week 236)
Replies: 29   Last Post: Aug 24, 2006 9:00 AM

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 Kevin Buzzard Posts: 1 Registered: 7/28/06
Re: This Week's Finds in Mathematical Physics (Week 236)
Posted: Jul 28, 2006 12:00 PM

John Baez <baez@math.removethis.ucr.andthis.edu> wrote:

[snip]

> At first these numbers seem to keep getting bigger! So, it seems
> shocking at first that they eventually reach zero. For example,
> if you start with the number 4, you get this Goodstein sequence:
>
> 4, 26, 41, 60, 41, 60, 83, 109, 139, 173, 211, 253, 299, 348, ...
>
> and apparently it takes about 3 x 10^{60605351} steps to reach zero!
> You can try examples yourself on this applet:
>
> 1) National Curve Bank, Goodstein's theorem,
> http://curvebank.calstatela.edu/goodstein/goodstein.htm

[note to jb: you wrote 41, 60 twice]

Although this number 3 x 10^{60605351} is the number quoted on the
website above, I did a back of an envelope calculation which
seemed to indicate that it took about (that number)^2 steps to
reach zero. In fact the website only claims that the sequence *increases*
until the 3 x 10^{60605351}th term, but it's not hard to check
that once the sequence has stopped increasing, it starts
decreasing very soon afterwards. Did I made a slip? I think
that the (2^n*2*n-2)'th term is zero, where n=24*2^24.

Kevin