Are there infinitely many primes of form $\underbrace{3\dots3}_n{}1$?

Question

The following numbers are prime:

$ 31 $
$ 331 $
$ 3331 $
$ 33331 $
$ 333331 $
$ 3333331 $
$ 33333331 $

Which made me think, is there something we can use to prove/disprove the statement that there are infinitely many primes of this form?

More precisely, can we prove/disprove that there are infinitely many primes of form:

$$\frac{10^{n+1}-7}{3}$$

This is prime for $n=1,2,3,4,5,6,7,17,39,49,59\dots$ since I tested all $n\le60$

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The only proofs for "infinitely many primes of form X" I know of are using the Dirichlet's theorem, but I don't see that it would be helpful in cases like this one.

Answer 1

A proof that there are infinite many primes of this form seems to be out of reach, but since the sequence is not increasing very fast and the $gcd$ of the numbers seems to be $1$, there are probably infinite many. The first few $n$ giving proven primes , calculated with PARI/GP are :

? for(n=1,500,m=(10^(n+1)-7)/3;if(isprime(m,2)==1,print1(n," ")))
1 2 3 4 5 6 7 17 39 49 59 77 100 150 318 381
?

You can also look up the known primes in this superb factorization database :

http://factordb.com/index.php?query=%2810%5E%28n%2B1%29-7%29%2F3&use=n&n=1&sent=Show&VP=on&VC=on&EV=on&OD=on&PR=on&PRP=on&U=on&perpage=20&format=1