That image prooves the point, it represent 3.2 billion bits of that algebraic number. f(N) = 1+1/4*(2*4^N+2*(16^N+1)^(1/2))^(1/2)/(2^N)^2 where N=32767
--I don't think my computer is powerful enough to handle that image. (I have no doubt yours is better than mine...) But I had no trouble reading Gosper's file http://gosper.org/plouffe.png which makes a nice change from his usual unreadability :) Anyhow, if by "proves the point" S.P. meant that the generating function explanation I gave cannot explain this, then I think you are wrong, rather it proves the gen.fn. DOES explain this. I think you may have fallen into the following trap. You correctly observed my Catalan-number example (which by the way was not originally due to me) stops seeming nonrandom quite soon, and the reason is the exponential growth of the Cat.numbers causing an example based upon the Cat.# generating function F(x) to "start looking random" after only about |log(x)| series coefficients, which is about the |log(x)|^2 bit in the binary number F(x). Note the SQUARING, which you may have forgotten to do. In your case, x is about 4^(-32767) or maybe 1/8 times this or something (I forget right now the exact definition I had to make for x, but only the order of magnitude matters so to hell with the 1/8 for the moment). The point is, log(x) = 65536 or so. Therefore, remembering the squaring, we expect nonrandomness out to about the 65536^2 bit, which is the 4 billionth bit. There is no precise point at which the nonrandomness ends, at least as far as I am concerned, but this is the neighborhood where it should fade out. So I repeat, your findings are compatible with my fairly simple gen.fn. explanation. I also repeat, that I described how to try to construct considerably-more-impressively-long nonrandom example numbers (and I see J.Propp mentioned a similar idea). You should try some of those suggestions with your number-compute+visualize software & see what you find. Hopefully it will confirm my predictions but I have not tried this computation myself and I am sure you have both better software & hardware than I; and using the ideas I outlined it should be possible using a more systematic exploration (which I have not done, and might not be completely trivial either) to optimize them. -- Warren D. Smith http://RangeVoting.org