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Dirichlet's Arithmetic Progression Theorem states that:

Given $a, b\in\mathbb{Z^+}$ with $(a,b)=1$, then $a+kb$ is prime for an infinite number of $k\in\mathbb{Z^+}.$

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 $j_k\in K_{a,b}$.

Is $c+j_k d$ prime an infinite number of times?

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    $\begingroup$ You are asking if $a+kb$ and $c+kd$ can be prime at the same time for infinitely many $k$ when $(a,b) = 1$ and $(c,d) = 1$. A simple counterexample is $k$ and $k+1$. You should look up Schinzel's Hypothesis H (qualitative conjecture) or the Bateman-Horn conjecture (quantitative conjecture) to see when a finite list of nonconstant polynomials $f_1(x), \ldots, f_r(x)$ with integer coefficients is expected to take on prime values infinitely often at the same time. A key "nonobvious" condition is that for each prime $p$, the product $f_1(x)\cdots f_r(x)$ is not identically 0 on $\mathbf Z/(p)$. $\endgroup$
    – KConrad
    Commented Dec 1, 2018 at 13:44
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    $\begingroup$ The special case of this conjecture when the $f_i(x)$ are all linear goes back to Dickson (1904). Look up Dickson's conjecture on Wikipedia. The title of Dickson's paper is similar to the title of your post: "A New Extension of Dirichlet's Theorem on Prime Numbers". $\endgroup$
    – KConrad
    Commented Dec 1, 2018 at 13:48

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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 $3+1\cdot\mathbb N$.

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