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Old 2020-11-13, 01:16   #2
Dr Sardonicus
 
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Feb 2017
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Quote:
Originally Posted by swishzzz View Post
Are problems like these remotely solvable using known number theoretic methods?
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2. Prove/disprove there are infinitely many primes whose decimal representation does not contain the digit 9. For large n there are roughly n/log(n) primes below it and roughly 9^(log(n)/log(10)) = 9^(log(n)/log(9) * log(9)/log(10)) = n^0.954 numbers that do not contain the digit 9. If we randomly picked n/log(n) numbers from 1..n for say set A and n^0.954 numbers from 1..n for set B, the probability of A and B having no common intersection is something like (1-1/log(n)) ^ (n^0.954) ~ exp(-n^0.954/log(n)) for large n assuming n^0.954 << n/log(n) which is pretty damn close to zero, but of course this isn't really a proof. However this also seems much weaker than one of Landau's problems which states that are infinitely many primes of the form n^2+1 so this one seems like it should be "easier".
I think this one is known... (Google Google) Here we go! Primes with restricted digits. PDF here.

So it is proven, but don't ask me to explain the proof...

Still thinking about first problem...
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