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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 $n\le60$

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.

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    $\begingroup$ This seems no easier than the question: are there infinitely many primes of the form $2^n-1$? And that one is a known open question. $\endgroup$ – GEdgar Apr 9 '17 at 9:55
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    $\begingroup$ This page will help: stdkmd.com/nrr/3/33331.htm $\endgroup$ – didgogns Apr 9 '17 at 9:57
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    $\begingroup$ Just for trivia, it was proved (in 1997) that there are infinitely many primes of the form $x^2+y^4$ also. It's called the Friedlander-Iwaniec theorem. However, no results come even close to the kind of result you are asking for. $\endgroup$ – астон вілла олоф мэллбэрг Apr 9 '17 at 10:31
  • $\begingroup$ Another open problem that looks even easier is whether $x^2+1$ is prime for infinite many positive integers $x$. More general, no polynomial with integer coefficients and degree greater than $1$ is known to produce infinite many primes. $\endgroup$ – Peter Apr 10 '17 at 12:35
  • $\begingroup$ See oeis.org/A055520 $\endgroup$ – Barry Cipra Apr 19 '17 at 18:43
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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 :

https://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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  • $\begingroup$ The database approves that $n=783,1731$ and $1917$ give the next primes coinciding with my pseudoprime-rotuine PARI/GP giving the same values. $\endgroup$ – Peter Apr 10 '17 at 10:54
  • $\begingroup$ Note that the isprime test is not pseudo $\endgroup$ – Hagen von Eitzen Apr 10 '17 at 10:58
  • $\begingroup$ @HagenvonEitzen isprime(n,2) proves the prime and isprime gives a "very probable" prime. But for numbers with more than $500$ digits it is generally quite slow. $\endgroup$ – Peter Apr 10 '17 at 11:01
  • $\begingroup$ Hm, according to what my PARi tells me in documentation, the result of isprime is always based on proven primality or compositeness, no matter if/what you have as second parameter (hough the proof may not always be output). Only ispseudoprime test for "very probable" prime .. $\endgroup$ – Hagen von Eitzen Apr 10 '17 at 12:11
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    $\begingroup$ Since this question won't get an answer with a proof unless there is some significant breakthrough, I'll accept this answer. $\endgroup$ – Vepir Jun 6 '17 at 16:14

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