Minimal Goldbach pairs in prime and twin-prime counting

Goldbach's Conjecture assumes that every even integer $2N \geq 4$ can be written as the sum of two primes $2N = p_i + p_j$, where $(p_i, p_j)$ is called a Goldbach pair. The minimal Goldbach pair is a pair $(p_i, p_j)$ such that $p_i$ is minimal and $p_j = 2N - p_i$ is also a prime. We define a function $F_{2N}(P)$ that counts the occurrences of $p_i = P$ in a set of minimal Goldbach pairs up to $2N$, where $P$ is a fixed prime number. In particular, the function $F_{2N}(P)$ provides the following identities in terms of prime counting $\pi(2N)$, twin-prime counting $\pi_2(2N)$ and cousin prime counting $\pi_4(2N)$

$$\pi(2N) = F_{2N+3}(3) + 1$$ $$\pi_2(2N) = F_{2N+3}(3) - F_{2N+5}(5)$$ $$\pi_4(2N) = F_{2N}(5) - F_{2N}(7)$$

Related MathOverflow question

• https://mathoverflow.net/q/54710/113033

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