Tim May said:

I stand by my point that no number or sequence can be proved to be random.

To expand a bit on Perry's arguments, the bottom line of all this research is that a claim regarding randomness can only be made *relative* to a particular system for specifying algorithms. In that sense, Tim's statement can be regarded to be correct, iff one assumes that a context (an algorithmic specification system) is not given. That is a huge qualifier, though, and not one to be taken for granted. A trivial example of this: pick some constant bitstring of length 32 or less. Call it K. Now look at the class of algorithms specifiable by the C code fragment printf("%x", K) --- i.e. print K as a hexadecimal number. Relative to that particular set of (one) algorithms, that value of K is trivially nonrandom, in the sense that the probability of of finding that bitstring produced by that class of algorithm is precisely 1. Next consider a program that computes an output by multiplying some input by two. The probability that the output will be K, given any possible (but unknown) input, is exactly zero if K happens to be odd. If K is not odd, then the probability depends on the distribution (randomness) of the inputs. Proceeding in this fashion, it becomes increasingly clear that the randomness of the output of an algorithm can only be measured relative to the properties of the class of algorithms being considered. Randomness in isolation is meaningless. The best sources of intuition regarding randomness usually derive from systems which shift the burden into an existing intuition on a slightly different subject. For instance, flipping a coin can be regarded as a random process in an intuitive sense, but only because it appeals to existing intuitions about equiprobablistic outcomes. Therefore one sees confused appeals to intuition about randomness, probability, entropy, or related ideas, in cryptography, quantum mechanics, information theory, statistical mechanics, philosophy (in regard to free will versus determinism versus randomness), etc, etc, but given Chaitin/Kolmogorov/et al, no intuition from any such subject should be taken at face value. There's more, but I'll pause to allow flames. :-) Doug