Comment by dataflow
2 days ago
> I'm going to be the nerd that points out that it has not been mathematically proven that pi contains every substring
Fascinating. Do you know if this has been proven about any interesting number (that wasn't explicitly constructed to make this true)?
You can use the number 0.{concatenation of all 1 bit strings ordered}{concatenation of all 2 bits strings ordered} = 0.0100011011…
Now an interesting problem would be to have the most “compact” number, where you can find all the strings of length n in a minimal number of digits of such number. It reminds me of the math problem solved by a 4chan user.
https://en.wikipedia.org/wiki/Normal_number has some examples and a bunch of info. Most of them are pretty artificial, but the concatenation of the primes one is... at least interesting, not obvious (to me) from doing that that it'd be normal.
Hmm, you are referring to rich numbers but pointing to normal numbers, so I will be the nerd who points out that every normal number is rich, but some rich numbers aren't normal.
https://en.wikipedia.org/wiki/Disjunctive_sequence
Oh nice, thanks, I didn't know that one.