# 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:

 $\displaystyle 11 + 199$ $=$ $\displaystyle 210$ $\displaystyle 13 + 197$ $=$ $\displaystyle 210$ $\displaystyle 17 + 193$ $=$ $\displaystyle 210$ $\displaystyle 19 + 191$ $=$ $\displaystyle 210$ $\displaystyle 29 + 181$ $=$ $\displaystyle 210$ $\displaystyle 31 + 179$ $=$ $\displaystyle 210$ $\displaystyle 37 + 173$ $=$ $\displaystyle 210$ $\displaystyle 43 + 167$ $=$ $\displaystyle 210$ $\displaystyle 47 + 163$ $=$ $\displaystyle 210$ $\displaystyle 53 + 157$ $=$ $\displaystyle 210$ $\displaystyle 59 + 151$ $=$ $\displaystyle 210$ $\displaystyle 61 + 149$ $=$ $\displaystyle 210$ $\displaystyle 71 + 139$ $=$ $\displaystyle 210$ $\displaystyle 73 + 137$ $=$ $\displaystyle 210$ $\displaystyle 79 + 131$ $=$ $\displaystyle 210$ $\displaystyle 83 + 127$ $=$ $\displaystyle 210$ $\displaystyle 97 + 113$ $=$ $\displaystyle 210$ $\displaystyle 101 + 109$ $=$ $\displaystyle 210$ $\displaystyle 103 + 107$ $=$ $\displaystyle 210$

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