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Let $m_{n}$ be the number we get from putting first $n$ primes together.

$$m_{1}=2$$ $$m_{2}=23$$ $$m_{3}=235$$ $$m_{4}=2357$$ $$m_{5}=235711$$ $$m_{6}=23571113$$ $$m_{7}=2357111317$$

and so on. Only primes i found on this sequence with Mathematica are $m_{1}=2, m_{2}=23, m_{4}=2357.$ For $5\leq n\leq 40$ is composite. Are there any more primes of the form $m_{n}$?

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  • $\begingroup$ Note that there is nothing specifically interesting about base $10$ in the context of this question (i.e., one could equally generate such sequence on any other base). $\endgroup$ – barak manos Feb 24 '16 at 16:14
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Yes there are others.

From the corresponding list in OEIS we can find the sublist of primes. Unfortunately, tough,

The next term is the 355-digit number 2357111317192329313741434753...677683691701709719 which is too large to include here.

There other interesting lists linked there, like this one, that is the list of "Numbers $n$ such that the concatenation of the first $n$ primes is a prime."

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These numbers are called Smarandache–Wellin numbers, and if they happen to be prime, they are called Smarandache–Wellin primes. There are many known primes of this form, and the first one after $2357$ is $m_{128}=235711\dots719$ (ending with prime $719$).

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The concatenation of the first $1,2,4,128,174,342,435,1429$ prime numbers are (probable) primes. And these are the only we know as of may 8th $2016$, M.Rodenkirch found no other primes up to the first $78498$ primes. Here http://mathworld.wolfram.com/IntegerSequencePrimes.html. These kind of primes has been called Smarandache-Wellin primes.

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