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Is this a known conjecture or theorem:

Among primes in the form $ax + b$ where $a, b$ are coprime (per Dirichlet's theorem on arithmetic progressions), $x$ is prime at least once.

Is this a known question? An answered question? Is it provable?

Thank you

EDIT:

I'm specifically looking at this: $Px + 2$ where $P$ is prime. $P$ and $2$ are coprime, obviously. $Px + 2$ is prime infinitely often; $x$ has to be odd; does $x$ have to be prime at least once?

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  • $\begingroup$ Based on the counter example, I would suggest that the conjecture be modified to not include any progression where $a\cdot b$ is an odd number. $\endgroup$ – abiessu Jun 2 '15 at 13:25
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It's not true. Take $a=7, b=1$. Then, if $x=2$, then $ax+b=15=3\cdot 5$; and if $x$ is an odd prime, then $ax+b$ is even and $>2$, so it's composite. So whenever $ax+b$ is prime, $x$ is composite.

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  • $\begingroup$ Good counter example. What if we apply the "simple" fix and say that in cases of odd $a,b$ we instead consider the series $2ax+b$? Or even drop the odd $a,b$ series from consideration altogether? $\endgroup$ – abiessu Jun 2 '15 at 13:21
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This was posted such long ago, but here is my answer. This seems to be a special case of Dickson's conjecture.

https://en.wikipedia.org/wiki/Dickson%27s_conjecture

Your question is pretty much asking for $x,y$ that are both prime, such that, $y-Px=2$ for some prime $P$. in other words, ${x,Px+2}$ is both prime. Which is exactly what the conjecture is about.

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