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 $\pi(n)$ ?
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1$\begingroup$ 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 $\pi(n).$ $\endgroup$– B. GoddardJul 22, 2019 at 10:40
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$\begingroup$ So you say that $\pi(n-2, n)$ < $\pi(n)$, and the difference is the number of twin prime pairs up to n. $\endgroup$– TilsightJul 22, 2019 at 11:53
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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.