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Math Help - Primes p 123456789

  1. #1
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    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)
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  2. #2
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    I suppose you're expected to use Dirichlet's theorem on arithmetic progressions? Take a look at the following sequence: a_n = 123456789 + 10^{10} n
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  3. #3
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    Is it possible to prove this using the fact that \sum_{p\in P}\frac{1}{p} diverges (where P=\{n\in\mathbb{N}: n \text{ is prime}\})?
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  4. #4
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    Quote Originally Posted by JoachimAgrell View Post
    Is it possible to prove this using the fact that \sum_{p\in P}\frac{1}{p} diverges (where 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 (123456789,10^{10})=1, Dirichlet's theorem on arithemtic progressions assures us that

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

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