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

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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 $\left[{105 \,.\,.\, 208}\right]$.



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