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If one day we finally prove the normality of $\pi $, would we be able to say that we have ourselves a sure-fire truly random number generator?

user132181
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  • Assuming this is the case, how would you generate a sequence of random numbers to use in any algorithm or application? If you want this sequence to be random, you'd need to start at a random position of the (say) decimal expansion of $\pi$, otherwise it is completely determininistic to any adversary.

    But you would indeed get a randomness expander: from any (e.g. constant) number of truly random bits, you could get arbitrarily many more.

     – Clement C. May 10 '14 at 19:34
  • It seems random to me in the sense that you can start at the 100th digit, and you won't know what the next digit is unless you calculate it. – user132181 May 10 '14 at 19:36
  • But if you want to use an algorithm such as a cryptographic protocol or anything requiring randomness for the purpose of "beating an adversary", then this is no good at all. – Clement C. May 10 '14 at 19:37
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    If I told you that a solution to $(x-1)(x^2+2x+2)=0$ was $x=1$, then the next solution would seem random, unless you can calculate it. – Emily May 10 '14 at 19:37
  • @Arkamis: "random" and "uniformly distributed" are not quite the same, though. A normal number would give some sort of stronger garantee, as as long as you haven't uncovered the next digit it looks uniformly distributed. – Clement C. May 10 '14 at 19:38
  • @ClementC. My comment had nothing to do with uniformity, but rather the fallacious assumption that difficulty in computation is a suitable alternative for randomness. In any case, I can easily construct a polynomial with uniformly distributed solutions. – Emily May 10 '14 at 19:39
  • @Arkamis -- indeed (actually, there are some connections between computational hardness and pseudorandom generators). I just wanted to point out that the OP seemed to use "ramdom" for "uniformly distributed" -- which is was normality is about. – Clement C. May 10 '14 at 19:41
  • Since a computer has finitely many states it can only produce eventually repeating sequences. So what do you mean by "generator" exactly? – WimC May 10 '14 at 19:55
  • http://en.wikipedia.org/wiki/Pseudorandom_generator#Definition The key is the computational part (indistinguishable from "true uniform randomness" by a family of computationally bounded algorithms). – Clement C. May 10 '14 at 20:00
  • Champernowne's constant is known to be normal. Its value is $0.1234567891011121314151617…$. Can it be used as a random number generator? – MJD Aug 19 '22 at 16:09

4 Answers4

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Not random. Never random. Evenly distributed, and pseudo-random yes. Pi is defined before you calculate it, so its not random, just unknowable before you calculate each digit. If you pick digits that have not been calculated or formulated (or just not know to you) then it is for all practical purposes it is random.

TL;DR version

No, but you can use it as such for simple applications

Asimov
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Tomorrow's lottery numbers are random. Yesterday's are not. Pi is like the latter.

If you're not convinced, we can take a bet on yesterday's numbers. :-)

vadim123
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A wonderful paper by Jacques Dutka I chanced to see in the 1970's put a million digit pi expansion through a rigorous set of tests for uniformity and unpredictability and showed it was significantly better that the best RNG at the time. Its utility was not limited by the "fact" that since it was entirely known to a million digits, and was an ontic fact (the certain relationship between a radius and circumference) it could not ITSELF be random.

castlestj
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  • Your answer could be improved with additional supporting information. Please [edit] to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center. – Community Feb 20 '24 at 04:03
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The definition of random is unpredictable. $\pi$'s digits have been calculated to billions of digits and they are all public information. You might be able to use a random range of the digits, but then you would need a separate method to select that range.

qwr
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