TITLE: A prime of the form $38111111\ldots$
QUESTION [15 upvotes]: Let $z(n)$ denote the number given by $38$ followed by $n 1$'s. 
What is the least number $n$, such that $z(n)$ is prime ?
With brute force, I checked up to $7000$ digits and did not find a prime.
Can anything useful be said about the possible prime factors which
could help to accelerate the search ?
For those, who prefer formulas 
$$z(n)=\frac{343*10^n-1}{9}$$

REPLY [2 votes]: The expression is congruent of 3.  
$343 \equiv 1 \pmod 3$  and    3|$10^n - 1$.