Dirichlet's Arithmetic Progression Theorem states that:



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> Given $a, b\in\mathbb{Z^+}$ with $(a,b)=1$, then $a+kb$ is prime for an infinite number of $k\in\mathbb{Z^+}.$
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For any given $a$ and $b$ let $K\_{a,b}=\{k\mid a+kb \text{ is prime}\}$.


Also consider another Dirichlet-Valid AP $c+jd$. Restrict $j$ to $j\_k\in K\_{a,b}$.



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> Is $c+j\_k d$ prime an infinite number of times?
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Extension of Dirichlet's Arithmetic Progression Theorem
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 $3+1\cdot\mathbb N$.


