Does $19,199,1999,\dotsc$ contain infinitely many prime numbers?

Are there infinitely many primes of the form $F_n =2\times10^n-1$? That is, does this sequence,

$$19,199,1999,\dotsc$$

contain infinitely many prime numbers?

I think about Dirichlet's theorem on arithmetic progressions, but the problem is difficult to start.

Edit: $F_n$ is prime for $n=1, 2, 3, 5, 7, 26, 27,\dotsc, 55347$ (A002957) and $n=1059002$ (Kamada's tables). $F_{n}$ for $n=6m+4$ is divisible by $7$.

Solution 1:

The current status of this conjecture is that it is not proven either way. It is similar to the mersenne primes conjecture, as well as the twin prime conjecture. At this point, the only infinitude proof concerning prime numbers of a specific kind gives an upper bound on the limit inferior of the difference between consecutive primes, which is somewhat vague and doesn't involve formulas as specific as yours. Ulam's spiral shows that there are many similar formulas, perhaps infinitely many, that often produce primes. Whether or not they do so infinitely often remains an open question.

It seems at this point that in order to prove such conjectures, we will need new tools and methodologies, as analytical number theory doesn't cut it and none of the other branches of mathematics have stepped in to pick up the slack. However, it is entirely possible that answers to these questions will begin to pop up within the next decade or even sooner. Once one such proof is assembled, others are sure to follow.