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For normality, see https://en.wikipedia.org/wiki/Normal_number. For random number/sequence, see https://en.wikipedia.org/wiki/Algorithmically_random_sequence.

Now, is there any number that is normal in every bases $b > 1$ except random numbers (or numbers expansion of which is algorithmically random sequence)?

I suspect, there is only normal number in some bases except random number, normality in every bases $b>1$ is a strong requirement to exclude computable numbers normal in some bases. In other word, is it possible that normality in every bases $b>1$ is contradicted theoretically?

Because computability of numbers is contradict to randomness. Normality in every bases $b>1$ may leads the number to be random one. So another question is: Could Normality in every bases $b>1$ pass the randomness test?

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    $\begingroup$ It is highly likely that nonrandom numbers like $\sqrt{2},\pi,$ and $e$ are normal in every base. Perhaps the question should be the following: can one give an explicit algorithm which provably produces a number that is normal in every base? $\endgroup$ Commented Aug 21, 2017 at 15:17
  • $\begingroup$ @RichardStanley I think it is impossible or very hard to give such algorithm. We know there are algorithms which convert number expansion from one bases into other, And some normal number in some bases are provably are not normal in other bases, for instances, Stoneham's constants. $\endgroup$ Commented Aug 21, 2017 at 15:29
  • $\begingroup$ Please see huffingtonpost.com/david-h-bailey/… $\endgroup$ Commented Aug 21, 2017 at 15:31
  • $\begingroup$ @RichardStanley yes, Your suggestion is simpler and more easy than my second question with the condition that there is a computable number in every bases $\endgroup$ Commented Aug 21, 2017 at 15:35

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Computable, absolutely normal numbers do actually exist. See

V. Becher, S. Figueira: An example of a computable absolutely normal number, Theoretical Computer Science 270 (2002), 947-958.

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  • $\begingroup$ So strange, there does exist a computable, absolutely normal number! $\endgroup$ Commented Aug 21, 2017 at 23:59
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    $\begingroup$ That's life :-) $\endgroup$ Commented Aug 22, 2017 at 7:45

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