It is Even, Not quite Odd

$38=37+1 = 35+3 = 33+5$ $= 31+7 = 29+9 = 27+11$ $=25+13 = 23+15 = 21+17 =19+19$ .

There is no combination where both the odd numbers are composite.

For numbers greater than 38, we have

$40 = 25 + 15$ , $42 = 27 + 15$ , $44 = 35 + 9$ , $46 = 25 + 21$ , $48 = 33 + 15$ , $50 = 35 + 15$ , $52 = 27 + 25$ , $54 = 27 + 27$ , $58 = 33 + 25$ , $60 = 45 + 15$ , $62 = 35 + 27$ , $64 = 39 + 25$ , $66 = 33 + 33$ , $68 = 35 + 33$ .

Each of the numbers on the RHS of the equations are either a multiple of 3 or of 5 or both.

Any even number greater than 68 can be written as

$n = 30k+n_1$ , where $n_1 \in \{40,42,\cdots,68\}$ and if $n_1 = c_1+c_2$ , then $n=30k+c_1+c_2$ . Now, $c_1$ is composite and is divisible by 3 or 5. This would mean that $30k+c_1$ is also composite since $30k$ would be divisible by 3 and 5.

Hence, any even number greater than 38 can be expressed as a sum of two odd composite numbers.