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Comment by 34679 | Hacker News Reader

Assuming that pi has an infinite number of digits, then the answer to "What is the longest known sequence that repeats in Pi?" is "The longest known sequence of Pi".

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This isn't true as you can build an infinite sequence that never repeats. An example sequence in binary is (the number of 0s between each 1 increases by 1 every time)

01001000100001...

Exactly. A number with the property that every sequence occurs is called a rich or disjunctive number - a number can be rich in s specific bases or rich for all bases we don't know whether pi is any if that. A number where every sequence occurs equally often (scaled to the length of the sequence) is called a normal number, which is an even stronger property.

While Pi is not proven to be a disjunctive number, nor the stronger condition of being normal, it is generally believed to be normal. That being said we don't have a proof of Pi being normal, nor disjunctive.

I am not familiar with how a proof of that would be constructed, as clearly numerical or computational measurements could never be conclusive.

Starting from the fourth digit,

01000100

But maybe I don’t understand your example

You didn't understand the original claim which is that because Pi has an infinite decimal representation, every subsequence of it has a repeat

Ahh, you’re saying something like every sequence containing at least 2 ones, does not repeat.

There may be some long repeats, but not all sequences repeat. Thanks!

Your example is governed by an arbitrary rule of your choosing. Pi is not.

That assumption is empirically supported, but is, as of now, conjectural and unproven. In fact, our ability to prove facts about the distribution of digits in pi is so weak that we can't even rule out statements such as "after a certain point, all digits of pi represented in base 10 are either 0 or 1."

The longest sequence of digits that eventually repeat (as defined in the article) is unboundedly large for any infinite sequence of digits, not just pi!

Using the pigeonhole principle, there must be at least one length N repeating string in the first N(N!+1) characters of any string.

Ha! Yea, that is true.. now if we only had a printout of it!

Post the IP of your printer, and I'll send you one.

1f:ca7:5ee5:b12d::ca7:eat5:b12d

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