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The only numbers that have been proven to be normal are artificially constructed ones such as Champernowne's Constant (and its variants), the Copeland-Erdos constant. No specific constant has been proven to be absolutely normal (in all bases).

It has been conjectured that all irrational algebraic numbers are normal, but in base 10? Does the conjecture state that they are normal in all integer bases (absolutely normal)?

How would one actually go about proving this?

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    $\begingroup$ Compare with this post. It is called "absolutely normal" then. $\endgroup$ Oct 20, 2019 at 19:38
  • $\begingroup$ There are some numbers which have been proven to be normal in every base. (In addition to that, almost all real numbers are absolutely normal.) $\endgroup$ Sep 9, 2020 at 15:25

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The conjecture is that all irrational algebraic numbers should be normal in all bases simultaneously, not just base $10$. (There is no mathematical reason why base $10$ is special.)

If we knew how to prove it we would have proved it! One circle of ideas can be explored starting from this paper of Bailey and misiurewicz.

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