# How may number pairs $(n - 2, n)$ are there, less than $n$, where $(n – 2)$ is prime and $n$ is composite?

I was wondering about number pairs, that differ by 2 on the natural numbers field. They can be twin primes, twin composites, and mixed. The mixed can be 2 types, either the first is prime, the second is composite, or the first is composite, the second is prime. I am specially interested in $$\pi(n-2, n)$$ , where $$(n – 2)$$ is prime and $$n$$ is composite. Are there some upper limits in terms of the number of primes, or can we say something about them that has some connection with $$\pi(n)$$ ?

• Your $n$ is doing double duty. There are no $n$'s less than $n$. Second, if $p$ is any odd prime, then $p+2$ is composite almost all the time. So the number of pairs you're looking for is $\pi(n).$ Commented Jul 22, 2019 at 10:40
• So you say that $\pi(n-2, n)$ < $\pi(n)$, and the difference is the number of twin prime pairs up to n. Commented Jul 22, 2019 at 11:53

all primes greater than 5 are 1,7,11,13,17,19,23, or 29 mod 30. Your $$n-2$$ prime $$n$$ composite cases are could be any of them mod 30 which means they have a maximum of $${4x\over 15}$$ cases up to x, assuming all cases are possible at once ( they aren't) this is also a very weak upper bound for the number of primes.