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2020-11-13, 01:16   #2
Dr Sardonicus

Feb 2017
Nowhere

43·101 Posts

Quote:
 Originally Posted by swishzzz Are problems like these remotely solvable using known number theoretic methods? 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...