Consider the arithmetic progressions $2+3\mathbb N$ and $1+5\mathbb N$ and observe that if $2+3k$ is prime, then $k$ is odd. In $1+5k$ is prime, then $k$ should be even. So, for any $k\in\mathbb N$ the numbers $2+3k$ and $1+5k$ cannot be simultaneously prime.

The same contradiction could be attained on the arithmetic progresions $2+1\cdot \mathbb N$ and $3+1\cdot\mathbb N$.

Consider the arithmetic progressions $2+3\mathbb N$ and $1+5\mathbb N$ and observe that if $2+3k$ is prime, then $k$ is odd. On the other hand, if $1+5k$ is prime, then $k$ should be even. So, for any $k\in\mathbb N$ the numbers $2+3k$ and $1+5k$ cannot be simultaneously prime.

The same contradiction could be attained on the arithmetic progresions $2+1\cdot \mathbb N$ and $3+1\cdot\mathbb N$.