Is it true for any $n=2p$ where $p$ is prime, that the number of twin primes less than $n$ approaches the number of prime pairs $(p_{1},p_{2})$ such that $p_{1} + p_{2} = n$?
For example, If we choose prime number 499, then $n=998$. For $n = 998$ there are 33 prime pairs and there are 35 twin primes less than 998.
If we choose larger values of $n=2p$, the number of prime pairs will converge to the number of twin primes less than $n$?
The Hardy-Littlewood conjecture would tell us that the number of primes $p$ less than $x$ such that $p+2$ is also a prime should asymptotically satisfy: $$\pi_2(x)\sim 2C_2\frac x{(\ln x)^2}$$
Where $C_2\approx .66$
On the other hand, standard heuristics tell us that the number of ways to express an even $x$ as the sum of two primes should be asymptotically $$2C_2\times \prod_{p\,|\,x,\;p≥3}\frac {p-1}{p-2}\times \frac x{(\ln x)^2}$$
See, e.g., this.
Thus, standard conjectures would tell us that, for large even $x$:
$$\frac {\text {the number of prime pairs} ≤ x}{\text {the number of ways to write} \;x\;\text{as the sum of two primes}}\sim \prod_{p\,|\,x,\;p≥3}\frac {p-2}{p-1}$$
To be sure, both conjectures are entirely unproven.