4 ms·
Not exactly the same thing, but related: I think everyone believes Pi is a Normal Number, but it isn’t proven. https://en.m.wikipedia.org/wiki/Normal_number h
by mcpherrinm 5y ago
Not exactly the same thing, but related: I think everyone believes Pi is a Normal Number, but it isn’t proven. https://en.m.wikipedia.org/wiki/Normal_number https://en.m.wikipedia.org/wiki/Normal_number
I think if we could prove Pi is normal, we could probably also prove your statement to be true (but I’m not sure about that)
- slopbop 5y agoNormal is actually a stronger claim than "contains any finite string as a substring". That normal numbers contain any finite string as a substring is a straightforward consequence of the infinite monkey theorem: https://en.wikipedia.org/wiki/Infinite_monkey_theorem https://en.wikipedia.org/wiki/Infinite_monkey_theorem To see that the converse does not always hold, you could take something like the Champernowne constant https://en.m.wikipedia.org/wiki/Champernowne_constant https://en.m.wikipedia.org/wiki/Champernowne_constant and pad it with 9s between each integer, ie .192939495969798999109911991299... so that you still contain every finite substring, but you have a >50% chance of a randomly selected digit being 9.
- erdos4d 5y agoI think there is a subtlety here that makes this fail. Normal does not imply that any finite substring exists, just that the probability of such a string existing is uniformly distributed within the space of possible values. There isn't any guarantee that you will actually see such a string, though you almost surely will.