A variation on Dirichlet's theorem on arithmetic progressions

Question

Dirichlet's theorem on arithmetic progressions says that if $a$ and $b$ are coprime, then $\{a+bL\}_{L \in \mathbb{N}}$ contains infinitely many prime numbers.

I wonder if the following claim is true:

If $a$ and $b$ are coprime, then $\{a+b^L\}_{L \in \mathbb{N}}$ contains infinitely many prime numbers.

Notice that if my claim is true, then Dirichlet's theorem is true.

Thank you very much!

Edit: After receiving a few helpful comments, perhaps I should change my question to: Is there an additional condition that will make my claim true (a condition on $a$ and $b$)?

Answer 1

Let $a=4, b=n^4\,$ with $\,n \gt 1\,$, then $a+b^L$ is not a prime for any $L \ge 1\,$ by Sophie Germain's:

$$a+b^L=4 + n^{4L}=(n^{2L}+2+2n^L)(n^{2L}+2-2n^L)$$

Answer 2

Here's an alternative view that doesn't try to answer your question but merely points out some observations you could make.

$\Bbb{Z}$ under the open set basis $U(a,b) = a + b\Bbb{N}$ forms a topological ring.

Fixing $a$ you can form a topology at least with $U(b) = a + b^{\Bbb{N}}$ since if $a + b^{\Bbb{N}} \cap a + c^{\Bbb{N}} \neq \varnothing$ say $x \in $ the intersection, then $a + b^n = x = a + c^m$ or $b^n = c^m$. There is a some work involved but you need to conclude that there is another basic open set $a + d^{\Bbb{N}}$ containing $x$ contained in the intersection.