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

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Theorem

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

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

This sequence is A118081 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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.

$\blacksquare$


Also see


Sources