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:
- $2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 26, 28, 32, 38$
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:
| \(\ds 18\) | \(=\) | \(\ds 9 + 9\) | ||||||||||||
| \(\ds 24\) | \(=\) | \(\ds 9 + 15\) | ||||||||||||
| \(\ds 30\) | \(=\) | \(\ds 21 + 9\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 15 + 15\) | ||||||||||||
| \(\ds 34\) | \(=\) | \(\ds 9 + 25\) | ||||||||||||
| \(\ds 36\) | \(=\) | \(\ds 9 + 27\) |
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.
| \(\ds 26 - 9\) | \(=\) | \(\ds 17\) | which is prime | |||||||||||
| \(\ds 26 - 15\) | \(=\) | \(\ds 11\) | which is prime | |||||||||||
| \(\ds 26 - 21\) | \(=\) | \(\ds 5\) | which is prime | |||||||||||
| \(\ds 26 - 25\) | \(=\) | \(\ds 1\) | which is not composite |
| \(\ds 28 - 9\) | \(=\) | \(\ds 19\) | which is prime | |||||||||||
| \(\ds 28 - 15\) | \(=\) | \(\ds 13\) | which is prime | |||||||||||
| \(\ds 28 - 21\) | \(=\) | \(\ds 7\) | which is prime | |||||||||||
| \(\ds 28 - 25\) | \(=\) | \(\ds 3\) | which is prime | |||||||||||
| \(\ds 28 - 27\) | \(=\) | \(\ds 1\) | which is not composite |
| \(\ds 32 - 9\) | \(=\) | \(\ds 32\) | which is prime | |||||||||||
| \(\ds 32 - 15\) | \(=\) | \(\ds 17\) | which is prime | |||||||||||
| \(\ds 32 - 21\) | \(=\) | \(\ds 11\) | which is prime | |||||||||||
| \(\ds 32 - 25\) | \(=\) | \(\ds 7\) | which is prime | |||||||||||
| \(\ds 32 - 27\) | \(=\) | \(\ds 5\) | 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
- 1989: Ronald E. Ruemmler: Problem 1328 (Math. Mag. Vol. 62, no. 4: p. 274) www.jstor.org/stable/2689772
- 1990: Solution to Problem 1328 (Math. Mag. Vol. 63, no. 4: pp. 273 – 280) www.jstor.org/stable/2690953
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $38$