Suppose you have N symbols (e.g. "1","2",...,"N" or "a","b",...,"$") and a string of these symbols (say, the first trillion digits of pi). Then does there exist a prime number whose N-ary representation contains that string of digits?



Prime numbers and strings of symbols
Yes. This follows from the strong version of [Bertrand's postulate](https://mathoverflow.net/questions/2724/strong-bertrand-postulate). For example, to see that there is a prime which contains the digits 314159, use the fact that there is a prime between 314159\*10^N and 
314159\*10^N\*(1.000001) for N sufficiently large.


