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

## Sources

- 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)

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $210$