Number of Representations as Sum of Two Primes

Theorem
The number of ways an integer $n$ can be represented as the sum of two primes is no greater than the number of primes in the interval $\left[{\dfrac n 2 \,.\,.\, n - 2}\right]$.

Proof
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.

That is, the number of primes in the interval $\left[{\dfrac n 2 \,.\,.\, n - 2}\right]$.

Hence the result.