Number of Representations as Sum of Two Primes
Let $n = p + q$ where $p \le q$ and both $p$ and $q$ are primes.
There can be no more different representations of $n$ as $p + q$ than there are the number of possible options for $q$.
As $q \ge p$, it follows that $q \ge \dfrac n 2$.
Note that as $p \ge 2$, it follows that $q \le n - 2$.
The number of possible values of $q$ is therefore equal to the number of primes between $\dfrac n 2$ and $n - 2$ inclusive.
Hence the result.
- July 1993: Jean-Marc Deshouillers, Andrew Granville, Wladyslaw Narkiewicz and Carl Pomerance: An Upper Bound in Goldbach's Problem (Math. Comp. Vol. 61, no. 203: 209 – 213)