Pi as the Mother Number

It has occurred to me during idle moments that pi, if it is normal and never
repeats, contains within itself all possible finite integer strings. Is
this interesting? It could be in the sense that pi therefore may contain
ALL RANDOM NUMBERS. What does it mean to say that pi contains all random
numbers? It means that you could generate any 100-digit number with your
best random number generator and if you searched long enough through the
expansion of pi you would find your 100-digit number. Once we find the
supposedly random number in pi it doesn't seem so random anymore. In fact,
in some cases, when we find the number in pi, we will have found a way to
actually "compress" the digit string and thus lower its Kolmogorov
complexity. For example if we find our 100-digit string starting at the
37th digit of pi, we can just say, go to the 37th digit of pi and print the
next 100 digits. But as a very astute mathematician pointed out to me, the
problem with using this principle to compress numbers is the problem of the
index, i.e. the index into pi. Typically it becomes larger than the random
number we are trying to compress. I have been working on half-baked notions
for how to compress indices into pi (eg. go to the 10^24 + 37th digit of pi
and print the next 100 digits) so as to increase the universe of
compressible, formerly "random" numbers. I think what I am doing is
attacking what I intuit are shaky intellectual foundations of the
mathematical treatment of randomness ;-) I wonder if these speculations
have any resonance in the "real world" of mathematics or if anybody has any
ideas on how to compress indexes into pi?


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