Positive Even Integers as Sum of 2 Composite Odd Integers in 2 Ways
Theorem
Let $n \in \Z_{>0}$ be a positive even integer.
Let $n$ be such that it cannot be expressed as the sum of $2$ odd positive composite integers in at least $2$ different ways.
Then $n$ belongs to the set:
- $\left\{ {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 38, 40, 44, 46, 52, 56, 62, 68}\right\}$
This sequence is A284788 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
The sequence of odd positive composite integers begins:
- $9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, \ldots$
which is more than we need for this proof.
This sequence is A071904 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Generating the possible sums of these proceeds as follows:
\(\ds 18\) | \(=\) | \(\ds 9 + 9\) | ||||||||||||
\(\ds 24\) | \(=\) | \(\ds 9 + 15\) | ||||||||||||
\(\ds 30\) | \(=\) | \(\ds 9 + 21\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15 + 15\) | ||||||||||||
\(\ds 34\) | \(=\) | \(\ds 9 + 25\) | ||||||||||||
\(\ds 36\) | \(=\) | \(\ds 9 + 27\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15 + 21\) | ||||||||||||
\(\ds 40\) | \(=\) | \(\ds 15 + 25\) | ||||||||||||
\(\ds 42\) | \(=\) | \(\ds 9 + 33\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15 + 27\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 21 + 21\) | ||||||||||||
\(\ds 44\) | \(=\) | \(\ds 9 + 35\) | ||||||||||||
\(\ds 46\) | \(=\) | \(\ds 21 + 25\) | ||||||||||||
\(\ds 48\) | \(=\) | \(\ds 9 + 39\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15 + 33\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 21 + 27\) | ||||||||||||
\(\ds 50\) | \(=\) | \(\ds 15 + 35\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 25 + 25\) | ||||||||||||
\(\ds 52\) | \(=\) | \(\ds 25 + 27\) | ||||||||||||
\(\ds 54\) | \(=\) | \(\ds 9 + 45\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15 + 39\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 21 + 33\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 27 + 27\) | ||||||||||||
\(\ds 56\) | \(=\) | \(\ds 21 + 35\) | ||||||||||||
\(\ds 58\) | \(=\) | \(\ds 9 + 49\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 25 + 33\) | ||||||||||||
\(\ds 60\) | \(=\) | \(\ds 9 + 51\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15 + 45\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 21 + 39\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 25 + 35\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 27 + 33\) | ||||||||||||
\(\ds 62\) | \(=\) | \(\ds 27 + 35\) | ||||||||||||
\(\ds 64\) | \(=\) | \(\ds 9 + 55\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15 + 49\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 25 + 39\) | ||||||||||||
\(\ds 66\) | \(=\) | \(\ds 9 + 57\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15 + 51\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 21 + 45\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 27 + 39\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 33 + 33\) | ||||||||||||
\(\ds 68\) | \(=\) | \(\ds 33 + 35\) |
This exhausts all possible sums of odd positive composite integers of $68$ or less.
It remains to be demonstrated that all positive even integers over $68$ can be so expressed in $2$ different ways.
Notice that one of $n - 15, n - 25, n - 35$ is divisible by $3$.
Similarly one of $n - 9, n - 15, n - 21, n - 27, n - 33$ is divisible by $5$.
For all above $n - k$ which is composite, since $k$ is composite, $\paren {n - k} + k$ is a way to express $n$ as the sum of $2$ odd positive composite integers.
Since $n > 68$, these two ways are distinct unless $n - 15$ is divisible by $15$.
In that case, $n - 45$ is also divisible by $15$, and $n - 45 \ne 15$.
Then $n = \paren {n - 15} + 15 = \paren {n - 45} + 45$ are two ways to express $n$ as the sum of $2$ odd positive composite integers.
$\blacksquare$
Historical Note
According to the footnote to the presentation of the solution to Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers in 1990: Solution to Problem 1328 (Math. Mag. Vol. 63, no. 4: pp. 273 – 280) www.jstor.org/stable/2690953, this particular result was deduced by Gary R. Minnich, who is reported to have been a student.
Sources
- 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): $68$