# 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.

The full list of such numbers is as follows:

- $1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 24, 30, 36, 42, 48, 60, 90, 210$

This sequence is A141340 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The list contains:

- $n \le 8$
- $n \le 18$ where $2 \divides n$
- $n \le 48$ where $2 \times 3 \divides n$
- $n \le 90$ where $2 \times 3 \times 5 \divides n$
- $210 = 2 \times 3 \times 5 \times 7$

## 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 $\closedint {\dfrac n 2} {n - 2}$.

The interval $\closedint {\dfrac {210} 2} {210 - 2}$ is $\closedint {105} {208}$.

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:

\(\ds 11 + 199\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 13 + 197\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 17 + 193\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 19 + 191\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 29 + 181\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 31 + 179\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 37 + 173\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 43 + 167\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 47 + 163\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 53 + 157\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 59 + 151\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 61 + 149\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 71 + 139\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 73 + 137\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 79 + 131\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 83 + 127\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 97 + 113\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 101 + 109\) | \(=\) | \(\ds 210\) | ||||||||||||

\(\ds 103 + 107\) | \(=\) | \(\ds 210\) |

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

## 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*: pp. 209 – 213) www.jstor.org/stable/2152947

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