[FredPret]

I'm guessing the answer to this question is no, but... for an irrational number like pi, can we guarantee that a certain digit sequence will occur somewhere in its infinite reaches?

That would be a wildly impractical but very fun way to encode information - if everybody had a couple of petabytes of pi on their harddrive one day, you could just send the starting and ending digit to communicate an arbitrary amount of information this way.

Of course you'd first have to search through the whole universe of digits to find a sequence that's just right.

[Arnavion]

Such numbers are called "normal numbers". Pi likely is one based on all the digits we've computed for it, but we don't have any way to prove a given number like pi is normal or not.

Relevant Numberphile video: https://www.youtube.com/watch?v=5TkIe60y2GI

[FredPret]

Very interesting, thank you. He starts talking about this at 8:20

[thfuran]

1.01001000100001... is infinite and non-repeating, but doesn't contain all substrings. A number that does is called a "normal" number, and it's not known whether pi is. It seems to be pretty normal though.

[kurisufag]

this in practice: https://github.com/ajeetdsouza/pifs