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May 11th, 2017, 03:32 AM   #1
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Square root of primes

Is the square root of a prime number normal? Or I guess more formally, if p is in the set of prime numbers, then is the square root of p normal?

I know the proof that the square root of a prime number is irrational, but irrationality and normality are NOT the same thing by a long shot. Not totally sure this is the right place for this, but this seems real analysis-y enough.

What about the square roots of other numbers that aren't perfect squares? Those are irrational too. Are they normal?

Maybe I should have put this under number theory... huh. I'm really not sure. Oh well.
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May 11th, 2017, 04:03 AM   #2
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Since the definition of "normal", I don't see how you could get an analytical proof of this. You'd have to look for counterexamples on a case by case basis. Perhaps if I had a clearer idea of how one quantifies the randomness of a sequence, I'd have more insight.
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May 11th, 2017, 07:53 AM   #3
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Proving anything at all to be normal

Now that you mention it, v8archie, I don't even know how one would go about proving anything at all to be normal, except for a sequence of digits which systemically generated every single sequence of digits. Does anybody have any clue how to do this? After googling around, I'm a bit stuck on this.
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May 11th, 2017, 09:16 AM   #4
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Although we can prove that almost all real numbers are normal, very few numbers are known to be normal.

Quote:
Originally Posted by Wiki
While a general proof can be given that almost all real numbers are normal (in the sense that the set of exceptions has Lebesgue measure zero), this proof is not constructive and only very few specific numbers have been shown to be normal. For example, Chaitin's constant is normal (and uncomputable). It is widely believed that the (computable) numbers √2, π, and e are normal, but a proof remains elusive.
https://en.wikipedia.org/wiki/Normal_number
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Last edited by Maschke; May 11th, 2017 at 09:20 AM.
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May 11th, 2017, 11:20 AM   #5
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I've just spotted an error in my earlier post which I've corrected below.
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Originally Posted by v8archie View Post
Since the definition of "normal" is statistical, I don't see how you could get an analytical proof of this. You'd have to look for counterexamples on a case by case basis. Perhaps if I had a clearer idea of how one quantifies the randomness of a sequence, I'd have more insight.
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