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.

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 $\closedint {105} {208}$.



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