TITLE: Proof that there are infinitely many prime numbers starting with a given digit string
QUESTION [26 upvotes]: To prove the following fact: given any sequence of digits in any base, eg 314159265358979323 base 10, there are infinitely many primes that start with these digits,eg when expressed in decimal they start with 314159265358979323. I think using a prime number theorem will be helpful . But I am still not getting a point of how to start the proof.

REPLY [6 votes]: This is a special case of the following fact, a simple and short proof of which appears on p. 4 of Disjunctive sequences: An overview, by Calude, Priese and Staiger (1997):

If $a_1, a_2, a_3, \dots$ , is a strictly
increasing infinite sequence of positive integers such that $\lim_{n \rightarrow \infty} (a_{n+1}/a_{n}) = 1$, then for any
positive integer $m$ and any integer base $b ≥ 2$, there is some (and hence infinitely many) 
$a_n$ whose expression in base $b$ starts with the expression
of $m$ in base $b$.

The desired result follows by taking $a_n$ to be the $n$th prime $p_n$, noting that the prime number theorem implies $p_n \sim n \ln n$, and hence that $\lim_{n \rightarrow \infty} (p_{n+1}/p_{n}) = 1$.