1. ## Primes p 123456789

Hi

Prove that there exist infinitely primes p such that the decimal representation contain the digits 123456789.

(Excuse-me my bad English, I don't speak that language)

2. I suppose you're expected to use Dirichlet's theorem on arithmetic progressions? Take a look at the following sequence: $\displaystyle a_n = 123456789 + 10^{10} n$

3. Is it possible to prove this using the fact that $\displaystyle \sum_{p\in P}\frac{1}{p}$ diverges (where $\displaystyle P=\{n\in\mathbb{N}: n \text{ is prime}\}$)?

4. Originally Posted by JoachimAgrell
Is it possible to prove this using the fact that $\displaystyle \sum_{p\in P}\frac{1}{p}$ diverges (where $\displaystyle P=\{n\in\mathbb{N}: n \text{ is prime}\}$)?
No. Dirichlet's theorem on arithmetic progressions is much stronger than this statement. In fact, since $\displaystyle (123456789,10^{10})=1$, Dirichlet's theorem on arithemtic progressions assures us that

$\displaystyle \sum_{p \equiv 123456789 \mod 10^{10}}\frac{1}{p}$

diverges. That $\displaystyle \sum_{p}\frac 1 p$ diverges is not enough!