# Primes of the form $4p+1$, with $p$ prime

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.

• If the result were proven, then the infinitude of such primes would no longer be open. – LSpice Feb 28 at 18:43
• @LSpice but the result was about expectations, not about existence. – Fedor Petrov Feb 28 at 21:49

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 $$p$$.
• Is there a precise reason for writing $(1-1/2)$ rather than $(1/2)$ directly? – Sylvain JULIEN Feb 28 at 19:13
• 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 $p$ – Fedor Petrov Feb 28 at 19:32