Infinite primes of the form $2kp+1$, $p$ is an odd prime and $k\in\mathbb{N}$

Question

How do I prove that there are infinitely many primes of the form $2kp+1$ such that $p$ is an odd prime and $k\in\mathbb{N}$. The hint in the book I am using suggests considering the number $(2q_1q_2\cdots q_r)^p-1$

I assumed a finite number of primes of the form $2k+1,$ namely $q_1,q_2,\cdots q_r$. We can see that the number $n={\underbrace{(2q_1q_2\cdots q_r)}_Q}^p-1\equiv1(\mod p)$.

Hence, $n$ is of the form $2kp+1$ such that $q_1,q_2\cdots q_r\nmid n$. $n$ can be factorized as $(Q-1)(1+Q+Q^2+Q^3\cdots+Q^{p-1})$. Now, both $(Q-1), (1+Q+Q^2+\cdots Q^{p-1})$ are of the form $2kp+1$. But how do I prove that one of them is either prime or has a new factor of the form $2kp+1$.

A similar question is asked here:
Prove that there are infinitely many primes of form 2kp+1
where p is an odd prime.
I could not find any useful solution out of it since it was too short.

Answer 1

Consider the arithmetic progressions

$$1,2p+1,2(2p)+1,3(2p)+1,\cdots$$

Since $1$ is relatively prime to $2p$, by Dirichlet's theorem on arithmetic progressions we know this sequence has infinitely many prime numbers in it.