# 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:

 $\displaystyle 18$ $=$ $\displaystyle 9 + 9$ $\displaystyle 24$ $=$ $\displaystyle 9 + 15$ $\displaystyle 30$ $=$ $\displaystyle 21 + 9$ $\displaystyle$ $=$ $\displaystyle 15 + 15$ $\displaystyle 34$ $=$ $\displaystyle 9 + 25$ $\displaystyle 36$ $=$ $\displaystyle 9 + 27$

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.

 $\displaystyle 26 - 9$ $=$ $\displaystyle 17$ which is prime $\displaystyle 26 - 15$ $=$ $\displaystyle 11$ which is prime $\displaystyle 26 - 21$ $=$ $\displaystyle 5$ which is prime $\displaystyle 26 - 25$ $=$ $\displaystyle 1$ which is not composite

 $\displaystyle 28 - 9$ $=$ $\displaystyle 19$ which is prime $\displaystyle 28 - 15$ $=$ $\displaystyle 13$ which is prime $\displaystyle 28 - 21$ $=$ $\displaystyle 7$ which is prime $\displaystyle 28 - 25$ $=$ $\displaystyle 3$ which is prime $\displaystyle 28 - 27$ $=$ $\displaystyle 1$ which is not composite

 $\displaystyle 32 - 9$ $=$ $\displaystyle 32$ which is prime $\displaystyle 32 - 15$ $=$ $\displaystyle 17$ which is prime $\displaystyle 32 - 21$ $=$ $\displaystyle 11$ which is prime $\displaystyle 32 - 25$ $=$ $\displaystyle 7$ which is prime $\displaystyle 32 - 27$ $=$ $\displaystyle 5$ which is prime

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.

$\blacksquare$