# Can we prove that infinite many primes begin with any given digitstring?

With Dirichlet's theorem, we can easily prove that infinite many primes end with a given digitstring with final digit $$1,3,7$$ or $$9$$.

Can we also prove that infinite many primes begin with a given digit-string with first digit non-zero ?

Intuively, this should be the case, but I wonder whether we can show it rigorously. I thought of prime gaps, but I am not sure whether the best proven prime gaps are sufficient.

Yes. All you need is a prime gap of the form $$p_{n+1}-p_n\lt (p_n)^\theta$$ for $$\theta\lt 1$$; this is well-known (Wikipedia suggests that Hoheisel was the first to prove a bound of this form). Once you've got that, it becomes a matter of simple math; let $$K$$ be the initial digit string, of length $$k=\lceil\log_{10}(K)\rceil$$. Then the difference between e.g. $$10^n\cdot K$$ and $$10^n\cdot(K+1)$$ is $$10^n$$ whereas we have $$10^n\cdot K\lt 10^{n+k}$$. Now just choose $$n$$ such that $$\theta\cdot (n+k)\lt n$$; then the gap between $$10^n\cdot K$$ and $$10^n\cdot (K+1)$$ is larger than the largest possible prime gap there.
Note that this proves even more that was asked: not just that there are infinitely many primes of the given form, but that given an initial digit string, then for every sufficiently large $$n$$ there's at least one $$n$$-digit prime beginning with that digit string.
By the Prime Number Theorem, for any fixed $$\epsilon>0$$, we have $$\pi((1+\epsilon)n)-\pi(n)\to\infty$$ as $$n\to\infty$$. In particular, for any number $$a$$ representing the desired initial digits, we will find many primes between $$a10^k$$ and $$(a+1)10^k$$ for all large enough $$k$$.