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

proof-writing prime-numbers recreational-mathematics

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

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