5
$\begingroup$

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.

$\endgroup$
5
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ So, the prime gaps are tight enough. Thank you (+1) $\endgroup$ – Peter Feb 21 '19 at 18:38
2
$\begingroup$

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 $k$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ It would be nice if you work out the claim about the prime counting function. $\endgroup$ – Peter Feb 21 '19 at 18:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.