
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*n2)'th term is zero, where n=24*2^24.
Kevin

