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

## 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:

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

## 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 $60$.

$\blacksquare$