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I am working on some problems related to primes $q$ of the form $q = 4p+1$ where $p$ is also prime. The infinitude of such primes is still open. But recently I found that If I were to count the number of such primes up to x, I should expect to find $Cx/(\log x)^2$ of them. Can anyone suggest any reference to this result? Or some references where I can find such a result with proof.

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    $\begingroup$ If the result were proven, then the infinitude of such primes would no longer be open. $\endgroup$ – LSpice Feb 28 at 18:43
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    $\begingroup$ @LSpice but the result was about expectations, not about existence. $\endgroup$ – Fedor Petrov Feb 28 at 21:49
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I think what you ask about is a partial case of Bateman–Horn conjecture for polynomials $f_1(x)=x,f_2(x)=4x+1$. The expected value of $C$ equals $$(1-1/2)(1-1/2)^{-2}\prod_{p>2}\left\{(1-2/p)(1-1/p)^{-2}\right\},$$ product is taken over primes. Roughly speaking, $1-1/2$ is the probability that $n$ is even and $1-2/p$ for odd prime $p$ is the probability that $n$ and $4n+1$ both are not divisible by $p$.

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  • $\begingroup$ Is there a precise reason for writing $(1-1/2)$ rather than $(1/2)$ directly? $\endgroup$ – Sylvain JULIEN Feb 28 at 19:13
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    $\begingroup$ in general we write $1-a/p$, where $a$ is the number of residues $n$ modulo $p$ for which at least of the numbers $f_i(n)$ is divisible by $p$ $\endgroup$ – Fedor Petrov Feb 28 at 19:32
  • $\begingroup$ Incidentally, is Bateman-Horn conjecture known to imply Goldbach's one? $\endgroup$ – Sylvain JULIEN Feb 28 at 21:11
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    $\begingroup$ @SylvainJULIEN I am afraid that no. If some large even number is not a sum of two primes, there still can be very large prime tuples of any prescribed form. At least it looks so. $\endgroup$ – Fedor Petrov Feb 28 at 21:48
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    $\begingroup$ @SylvainJULIEN: On a strictly logical level Bateman-Horn does not seem to imply Goldbach. However, the methods we know do not see any difference between these problems, so it is quite likely that if one of prime twins, Goldbach, or Sophie-Germain primes is solved, then all of them are solved within the following year. $\endgroup$ – Jan-Christoph Schlage-Puchta Mar 2 at 11:46

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