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Is there such $p$ that for all $q$, $4pq+2p+1$ isn't prime?

As $4pq+2p+1=2p(2q+1)+1$, I guess the problem can be restated as "is there such an even number, that its product with any odd number plus one isn't prime?", but that's as far as I can go.

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If $2p + 1$ and $4p$ are coprime, then there exists at least one (infinitely many, actually) $q$ such that $4pq + 2p + 1$ is a prime number. This is Dirichlet's theorem.

Now, we just have to prove that $2p + 1$ and $4p$ are always coprime. To do this, consider $d$, the GCD of the two. Then it can be deduced that $d = 1$ or $d=2$. I will leave this deduction to you.

But $2\not| (2p+1)$. So their GCD must be equal to $1$, i.e. they must be coprime.

Hence, we see that no such positive integer $p$ exists.

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