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

## Contents

## 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

- 1989: Ronald E. Ruemmler:
*Problem 1328*(*Math. Mag.***Vol. 62**,*no. 4*: 274) www.jstor.org/stable/2689772

- 1990:
*Solution to Problem 1328*(*Math. Mag.***Vol. 63**,*no. 4*: 273 – 280) www.jstor.org/stable/2690953

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $38$