Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways

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Theorem

The sequence of positive integers $n$ which are the smallest such that they are the sum of $2$ odd primes in $k$ different ways begins as follows:

$k$ $n$
$1$ $6$
$2$ $10$
$3$ $22$
$4$ $34$
$5$ $48$
$6$ $60$

This sequence is A001172 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\displaystyle 6\) \(=\) \(\displaystyle 3 + 3\) $1$ way


\(\displaystyle 8\) \(=\) \(\displaystyle 5 + 3\) $1$ way


\(\displaystyle 10\) \(=\) \(\displaystyle 7 + 3\) $2$ ways
\(\displaystyle \) \(=\) \(\displaystyle 5 + 5\)


\(\displaystyle 12\) \(=\) \(\displaystyle 7 + 5\) $1$ way


\(\displaystyle 14\) \(=\) \(\displaystyle 11 + 3\) $2$ ways
\(\displaystyle \) \(=\) \(\displaystyle 7 + 7\)


\(\displaystyle 16\) \(=\) \(\displaystyle 13 + 3\) $2$ ways
\(\displaystyle \) \(=\) \(\displaystyle 11 + 5\)


\(\displaystyle 18\) \(=\) \(\displaystyle 13 + 5\) $2$ ways
\(\displaystyle \) \(=\) \(\displaystyle 11 + 7\)


\(\displaystyle 20\) \(=\) \(\displaystyle 17 + 3\) $2$ ways
\(\displaystyle \) \(=\) \(\displaystyle 13 + 7\)


\(\displaystyle 22\) \(=\) \(\displaystyle 19 + 3\) $3$ ways
\(\displaystyle \) \(=\) \(\displaystyle 17 + 5\)
\(\displaystyle \) \(=\) \(\displaystyle 11 + 11\)


\(\displaystyle 24\) \(=\) \(\displaystyle 19 + 5\) $2$ ways
\(\displaystyle \) \(=\) \(\displaystyle 13 + 11\)


\(\displaystyle 26\) \(=\) \(\displaystyle 23 + 3\) $3$ ways
\(\displaystyle \) \(=\) \(\displaystyle 19 + 7\)
\(\displaystyle \) \(=\) \(\displaystyle 13 + 13\)


\(\displaystyle 28\) \(=\) \(\displaystyle 23 + 5\) $2$ ways
\(\displaystyle \) \(=\) \(\displaystyle 17 + 11\)


\(\displaystyle 30\) \(=\) \(\displaystyle 23 + 7\) $3$ ways
\(\displaystyle \) \(=\) \(\displaystyle 19 + 11\)
\(\displaystyle \) \(=\) \(\displaystyle 17 + 13\)


\(\displaystyle 32\) \(=\) \(\displaystyle 29 + 3\) $2$ ways
\(\displaystyle \) \(=\) \(\displaystyle 19 + 13\)


\(\displaystyle 34\) \(=\) \(\displaystyle 31 + 3\) $4$ ways
\(\displaystyle \) \(=\) \(\displaystyle 29 + 5\)
\(\displaystyle \) \(=\) \(\displaystyle 23 + 11\)
\(\displaystyle \) \(=\) \(\displaystyle 17 + 17\)


\(\displaystyle 36\) \(=\) \(\displaystyle 31 + 5\) $4$ ways
\(\displaystyle \) \(=\) \(\displaystyle 29 + 7\)
\(\displaystyle \) \(=\) \(\displaystyle 23 + 13\)
\(\displaystyle \) \(=\) \(\displaystyle 19 + 17\)


\(\displaystyle 38\) \(=\) \(\displaystyle 31 + 7\) $2$ ways
\(\displaystyle \) \(=\) \(\displaystyle 19 + 19\)


\(\displaystyle 40\) \(=\) \(\displaystyle 37 + 3\) $3$ ways
\(\displaystyle \) \(=\) \(\displaystyle 29 + 11\)
\(\displaystyle \) \(=\) \(\displaystyle 23 + 17\)


\(\displaystyle 42\) \(=\) \(\displaystyle 37 + 5\) $4$ ways
\(\displaystyle \) \(=\) \(\displaystyle 31 + 11\)
\(\displaystyle \) \(=\) \(\displaystyle 29 + 13\)
\(\displaystyle \) \(=\) \(\displaystyle 23 + 19\)


\(\displaystyle 44\) \(=\) \(\displaystyle 41 + 3\) $3$ ways
\(\displaystyle \) \(=\) \(\displaystyle 37 + 7\)
\(\displaystyle \) \(=\) \(\displaystyle 31 + 13\)


\(\displaystyle 46\) \(=\) \(\displaystyle 43 + 3\) $4$ ways
\(\displaystyle \) \(=\) \(\displaystyle 41 + 5\)
\(\displaystyle \) \(=\) \(\displaystyle 29 + 17\)
\(\displaystyle \) \(=\) \(\displaystyle 23 + 23\)


\(\displaystyle 48\) \(=\) \(\displaystyle 43 + 5\) $5$ ways
\(\displaystyle \) \(=\) \(\displaystyle 41 + 7\)
\(\displaystyle \) \(=\) \(\displaystyle 37 + 11\)
\(\displaystyle \) \(=\) \(\displaystyle 31 + 17\)
\(\displaystyle \) \(=\) \(\displaystyle 29 + 19\)


From Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways, the smallest positive integer which is the sum of $2$ odd primes in $6$ different ways is $60$.

$\blacksquare$


Sources