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 $2 \times 3 \times 5 \divides n$
$210 = 2 \times 3 \times 5 \times 7$


Sufficient Condition

Let $n$ be a (strictly) positive integer.

Suppose we have for some positive integer $k$:

$p_k \# \divides n$
$n \le 2 {p_{k + 1} }^2$

where $p_k \#$ denotes the product of the first $k$ primes.


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

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 $\closedint {105} {208}$.


This theorem requires a proof.
In particular: It remains to be shown that $210$ is the largest number that can be represented by the maximum of these.
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Sources

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