Integers whose Number of Representations as Sum of Two Primes is Maximum

Theorem
$210$ is the largest integer which can be represented as the sum of two primes in the maximum number of ways.

Proof
From Number of Representations as Sum of Two Primes, 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 $\left[{\dfrac n 2 \,.\,.\, n - 2}\right]$.

The interval $\left[{\dfrac {210} 2 \,.\,.\, 210 - 2}\right]$ is $\left[{105 \,.\,.\, 208}\right]$.

The primes in this interval can be enumerated:


 * $107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199$

It can be seen there are exactly $19$ of them.

We have:

and as can be seen, there are $19$ such representations, one for each prime in $\left[{105 \,.\,.\, 208}\right]$.