Great comments on yesterday’s post!
In response to Manu’s comment, it doesn’t really make a difference. The base 96 encoding is equivalent to the same problem posed in binary encoding, except that one base 96 digit represents between 6 and 7 binary digits, since log2 of 96 is about 6.58496. So it’s really just a gloss on a problem that has already been extensively studied: The probability of finding any particular binary string within a random infinite binary sequence. You just apply that 6.58… constant afterward.
In response to Andy’s comment, you will indeed find every valid mathematical proof. But you won’t find a valid proof that Pi is rational, because it can be proven that such a proof cannot exist in any Universe.
In response to the third comment, I totally agree!!! To quote an old TV ad for Seven Up soda:
It’s the nothing that makes us something
It’s what we miss that hits the mark
It’s what’s left out that leaves us in
It’s the light shining over the dark
Those words are also written somewhere in the digital expansion of every transcendental number. And so is every other TV ad jingle that could ever be written, God help us all.
Sorry for being pedantic, but you’re talking about “normal” numbers, not transcendental.
The counterexample that comes to mind is something like Liouville’s number:
L= 0.110001000000000000000001 in which the nth digit after the decimal point is 1 if is the factorial of a positive integer and 0 otherwise. (Quoting from https://en.wikipedia.org/wiki/Liouville_number )
L is clearly is missing most patterns of ones and zeroes.
Pi is only conjectured to be normal (in all bases), so in theory there could be some sequences that it misses. Same for e I think. https://en.wikipedia.org/wiki/Normal_number
Thanks for the correction!