Positive Even Integers as Sum of 2 Composite Odd Integers in 2 Ways

From ProofWiki
Jump to navigation Jump to search

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