Is Dirichlet's Theorem lenient? Regarding Dirichlet's Theorem on primes in an arithmetical progression, is it true that for each real number $w$ there exist relatively prime positive integers $a$ and $d$ such that for the sequence whose first term is $a$ and whose constant difference is $d$, the first prime occurring in the sequence
has a subscript (as a term within the arithmetical sequence) greater than $w$?

Let $a=N!+2$, where $N$ is an integer $\gt |w|+3$, and let $d=1$. 
