Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers

Theorem
The positive even integers which cannot be expressed as the sum of $2$ composite odd numbers are:


 * $2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 26, 28, 32, 38$

Proof
The smallest composite odd numbers are $9$ and $15$, so trivially $2$ to $16$ and $20$ to $22$ cannot be expressed as the sum of $2$ composite odd numbers.

We have:

It remains to investigate $26, 28$ and $32$.

This will be done by progressively subtracting smaller composite odd numbers from them, and noting that the difference is not composite.

It remains to be demonstrated that all even integers greater than $38$ can be expressed as the sum of $2$ composite odd numbers.

We note that $9 + 6 k$ is odd and a multiple of $3$.

Numbers $18$ and greater of the form $6 n$ can be expressed as:
 * $\left({9 + 6 k}\right) + 9$

Numbers $34$ and greater of the form $6 n + 4$ can be expressed as:
 * $\left({9 + 6 k}\right) + 25$

Numbers $44$ and greater of the form $6 n + 2$ can be expressed as:
 * $\left({9 + 6 k}\right) + 35$

We have that $40$ and $42$ are of the form $6 n + 4$ and $6 n$ respectively.

Hence all even integers greater than $38$ are accounted for.

Also see

 * Positive Even Integers as Sum of 2 Composite Odd Integers in 2 Ways


 * Largest Even Integer not expressible as Sum of 2 k Odd Composite Integers