Take a bunch of primes: 2 3 5 7 ..

Concatenate them 2357. Either it is a prime or not. If it is a prime then…

Now is there a name for this? What is Smarandache prime? Are there infinitude of them? What can they tell about primes themselves?

But most interestingly. Suppose we take 2357111317192329… Then when will the first pattern appear such as it would read

2357111317192329

OEIS

Numbers n such that the concatenation of the first n primes (A019518) is a prime.

n a(n)

1 2

2 3

3 7

4 719

5 1033

6 2297

7 3037

8 11927

[2,3,7,719,1033,2297,3037,11927]

So is 23771910332297303711927 a prime? Yeah.

So it only remains to ask if an ouroborian [sic] capture is possible or not. Can the beginning of the sequence come back to itself at it’s tail?

Are there infinite of such beast?

Can Smarandahce numbers help us prove RH? Why can’t we prove that there are infinite of such Smarandache or not?

[Puzzle]

How to conceive a large Smarandache prime number? Can a prime number be so tall that once one finds a suitable totem one sees infinite extension to both ends and everything becomes meaningless.