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Whether or not $\pi$ is a normal number has yet to be determined. That is to say, we do not know for sure that its base expansion’s infinite sequence of digits is distributed uniformly.

Speaking hypothetically, if it was shown that $\pi$ was a normal number would this alter our views on number theory as a whole? If so, to what extent? What other mathematical claims could we deduce from this fact?

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  • $\begingroup$ what is Pi? do you mean $\pi$? $\endgroup$
    – johnny09
    Apr 24, 2019 at 3:03
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    $\begingroup$ You could prove that $\pi+1$ is normal. $\endgroup$ Apr 24, 2019 at 3:09
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    $\begingroup$ I guess not but, of course, I could be wrong. Normality depends on how we choose to write numbers. Most of the other properties studied in number theory do not. For example, whether a number is prime, irrational, or transcendental don't. We might expect alien mathematicians to know that $\pi$ is irrational and transcendental but they might even have asked the question of whether it is normal. $\endgroup$
    – badjohn
    Apr 24, 2019 at 8:46
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    $\begingroup$ I cannot imagine of any effect it would have if $\pi$ would proved to be normal , apart from the big suprprise for many mathematicians that it could be shown. $\endgroup$
    – Peter
    Apr 24, 2019 at 18:01
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    $\begingroup$ It wouldn't mean anything at all; the main reason to try proving things like this is to see what new ideas the proof contains. What would be much more interesting is if someone proved that $\pi$ isn't normal; that would be really astonishing. $\endgroup$ Apr 24, 2019 at 22:11

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