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Similar to: Does there exist a prime that is only consecutive digits starting from 1?

Let $b_n=\overline{a_1a_2a_3\dots a_n}$ and $a_n=n$. For example $b_{11}= 1234567891011$. I have a couple of questions about the primes in the sequence $b_n$.

  1. What is the lowest number $n$ such that $b_n$ is prime (if there is is an $n$)?
  2. Are there infintely many primes in the sequence $b_n$?
  3. Do these primes have a name?

The first question was an excercise in Clifford Pickover's A Passion For Mathematics. I don't know if there is a way to find the answer to the first question without using a computer.

Like in Does there exist a prime that is only consecutive digits starting from 1?, $n$ cannot be an even number (or else $b_n$ would be even). Thanks.

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  • $\begingroup$ If noone finds an answer, the recursion is $$b_{n+1} = 10^{\lfloor \log_{10}(n+1) \rfloor} b_n + n + 1$$ So you can write a script to test, similar to the linked question. $\endgroup$ – AlexR Sep 30 '14 at 7:15
  • $\begingroup$ @AlexR Thanks, I still hope someone finds an answer though. $\endgroup$ – Joao Sep 30 '14 at 7:16
  • $\begingroup$ You should add an example just to make sure people understand the notation. For example, I believe that $b_{11}$ is the thirteen-digit number 1,234,567,891,011; confirming that would be helpful. $\endgroup$ – Greg Martin Sep 30 '14 at 7:23
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    $\begingroup$ None of $b_2,\dots,b_{2000}$ is prime. But a probabilistic heuristic suggests that infinitely many of the $b_n$ should be prime, although they will be extremely rare. $\endgroup$ – Greg Martin Sep 30 '14 at 7:23

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