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Comment by schoen | Hacker News Reader

There is a dead post in this thread talking about what if "at 101T digits we find an infinite series of 000000000".

I just wanted to point out that this is impossible, because in that case pi would be a rational number. However, it's been proven to be a transcendental number, which is a stronger condition than just being irrational. So, it's not possible that pi ends with any infinitely repeating sequence.

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Fun fact: Mahler famously showed that if p/q is a rational number then |\pi - p/q| > 1/q^42. So we can't have a string of 4242T zeros after first 101T digits :-) The exponent 42 has since been improved by a series of works to 7.1032 - https://arxiv.org/abs/1912.06345

You can't have a string of 4242T zeroes immediately after the first 101T digits but presumably you could have them sometime later (say after another 10000T digits)

I seem to remember reading a sci-fi story quite a long time ago where the characters discover a message embedded in pi but only after many digits were calculated.

I think the implication was something about simulations or maybe instructions on how to build a device. Can't quite remember the details.

That sounds like Contact by Carl Sagan

In Northern Exposure season 4 episode 3 ("Nothing's Perfect"), a grad student comes to Cicely to study Pi.

"I just have this feeling if I take pi, well past all this static, take pi to 10 million, 20 million digits, that I'll find something really incredible. Not just a pattern, not just an order, but a sign. A mathematical sign."

https://moosechick.com/43.html

It is almost certain that pi does contain sequences of zeroes of arbitrary length, but that following those zeroes are other numbers.

Isn’t that still just a conjecture that pi’s digits are random-ish?

If you're talking about whether pi is 'normal', it's more than 'just a conjecture' but not something that's been proved.

https://en.wikipedia.org/wiki/Normal_number

Wtf is there between a conjecture and something which has been proven?

Fortunately, Philip J. Davis wrote a paper on this exact topic in 1985: https://www.gwern.net/docs/www/www.maa.org/9230a8f2614ddd74b...

He listed at least 21 states between "conjecture" and "proof".

Intermediate results, on which eventual proofs often end up relying.

But more weirdly, it's possible to find any string of digits within the digits of pi. So there are going to be quite a few zeros in there, followed by ever more non-zeros.

Transcendental numbers are truly awesome.

Small note, this is believed to be true but has not been proven.

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