Help on an application of Dirichlet's theorem for primes in progression

Question

Suppose that I have an infinite sequence of positive integers

$$a_1,\ldots,a_m,\ldots$$

with the following recursion

$$a_{m+1} -a_m =b(m+1)$$

So that

$$a_{m+1} =b(m+1) +a_m$$

Suppose further that I know that

$$(b,a_m)=1$$

Would Dirichlet's Theorem tell us that there are infinitely many primes of the form $a_{m+1} $

Answer 1

Take $b=1$. Then, (with $a_1=1$), $a_{m+1}=1+2+\cdots +(m+1)$. This is non-prime for $m\gt 1$, since it is $\frac{(m+1)(m+2)}{2}$.