Find all $n$ such that $6^{n}+19$ is prime.
By looking at the small cases, we realize that all these numbers are divisible by 5. Let's prove it:

$$
6^{n}+19 \equiv 1^{n}+19 \equiv 1+19 \equiv 0 \quad(\bmod 5)
$$

![](https://cdn.mathpix.com/cropped/2024_05_10_6d86e9a87432417d5c97g-200.jpg?height=55&width=1450&top_left_y=538&top_left_x=180)