Simon Plouffe writes: << What could anyone do to test a big super computer like [Hitachi's] ? It would be perhaps interesting to see if we could use the so called Earth Simulator (35 teraflop/s) to compute pi once.
This makes a lot of sense -- to use digits of pi to test supercomputers. (Though it seems clear this is only one of about aleph_1 different tests that would all serve equally well.) (Note: My remark about pi's digits possibly containing a message from another planet was a reference to part of the plot of Carl Sagan's sci-fi novel "Contact" -- which was a fun book, despite this nonsense.) Then, am I right that it has no particular mathematical value? (My feeling is that there *must* be some kind of pattern in the digits of any basic constant C, since two straighforward algorithms (computation of C and computation of its, say, decimal digits) would seem to necessarily have some kind of relationship. I confess that finding such a relationship, though perhaps not surprising to me, would be interesting, especially if the base of the number system were less random than 10 -- I'd prefer 2. So the issue of understanding {3^n} expressed in binary seems in fact a very interesting question. N.B. My favorite base of all, which like continued fractions has infinitely many distinct "digits", is factorial base (which I think many math-funners have seen). That is, for an positive integer n write n = a_k * k! + a_(k-1) * (k-1)! + ... + a_1 * 1! for integers 0 <= a_k <= k. And for a fractional part x, i.e., 0 <= x < 1, write it as x = n_1 / 2! + n_2 / 3! + ... + n_k / k! + ... where 0 <= n_k <= k for all k. (This works for x = 1, but in that case why bother?) I'd think the best chance of finding a pattern for pi would be in its continued fraction, or in factorial base. (Maybe for the cleanest representation, one should represent 1/pi, instead of pi, in factorial base.) --Dan