"More"-Cam [Large Partial Quotient in ContinuedFraction]

[Hans Havermann]

on 6/15/01 1:09 PM, Richard Guy wrote:

> For my next act (copied to math-fun as well) I tried
> A058631, and found the world's largest partial quotient!
> 
> My deep method was to set \precision=1000 in PARI,
> then type in cf(0.01101001...), the Thue-Morse
> sequence, using C^K and C^Y to copy chunks of sequence.
> 
> You can check how far and how correctly I went, but the
> results agree as far as the webcam shows.
> 
> The immense p.q. occurs somewhere around where I'd
> expect the precision to expire, but how did it happen?

The *exact* continued fraction of your approximation *ends* with the term
prior to your huge quotient. The precision of your approximation is only
about 638. By setting PARI's precision to 1000, I think (perhaps) that you
are introducing some 362 non-existent zeros after the number and your huge
partial quotient reflects this.

There are, of course, *real* large partial quotients in the continued
fraction expansion of Champernowne. The term at index 4838, for example,
contains 57111096 digits and I expect the number of digits of the term at
index 13522, if ever computed, to exceed this by an order of magnitude.

The winner, likely, is the simple continued fraction of BesselI[1,
2]/BesselI[0, 2] (0.697774657964...), which I understand contains
"googolplex" somewhere in its sequence. :)