5 ms·
Dumb question, but is any arbitrary string of digits with length N, somewhere in pi?
by mcbuilder 5y ago
Dumb question, but is any arbitrary string of digits with length N, somewhere in pi?
- deleted 5y ago[deleted]
- mcpherrinm 5y agoNot 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.
- 6gvONxR4sf7o 5y agoAt the very least, it doesn’t follow from the fact that it’s infinite and nonrepeating.
- phantomherman 5y agoI'm struggling to wrap my head around this: why doesn't it follow from that fact?
- kundor 5y agoYou can have infinite and nonrepeating sequences which don't contain every possible subsequence. For instance, 1010010001000... (where there is one more zero each time) never repeats itself, but it never even has the digit 2 in it.
- phantomherman 5y agoAha, makes sense, thank you!
- mateo1 5y agoIt's possible that the answer is no.
- erdos4d 5y agoNot dumb at all, an area of open research actually. The answer is that this is not known and you will be math-famous if you figure it out.