Largest Even Integer not expressible as Sum of 2 k Odd Composite Integers

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Theorem

Let $k \in \Z_{>0}$ be a (strictly) positive integer.

The largest even integer which cannot be expressed as the sum of $2 k$ odd positive composite integers is $18 k + 20$.


Proof

Let $n$ be an even integer greater than $18 k + 20$.

Then $n - 9 \paren {2 k - 2}$ is an even integer greater than $18 k + 20 - 9 \paren {2 k - 2} = 38$.

By Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers, every even integer greater than $38$ can be expressed as the sum of $2$ odd positive composite integers.

Thus $n - 9 \paren {2 k - 2}$ can be expressed as the sum of $2$ odd positive composite integers.

So, let $a$ and $b$ be odd composite integers such that $a + b = n - 9 \paren {2 k - 2}$.

Then:

$9 \paren {2 k - 2} + a + b = n$

This is an expression for $n$ as the sum of $2 k$ odd positive composite integers, in which $2 k - 2$ of them are occurrences of $9$.

Thus such an expression can always be found for $n > 18 k + 20$.

$\Box$


It remains to be shown that $18 k + 20$ is not expressible as the sum of $2 k$ odd positive composite integers.

The $k = 1$ case is demonstrated in Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers.


Now suppose $k \ge 2$.

The two smallest odd positive composite integers are $9$ and $15$.

Suppose $18 k + 20$ is expressible as the sum of $2 k$ odd positive composite integers.

Then at least $2 k - 3$ of them are $9$'s, because:

$9 \paren {2 k - 4} + 4 \times 15 = 18 k + 24 > 18 k + 20$

Then the problem reduces to finding an expression of $18 k + 20 - 9 \paren {2 k - 3} = 47$ as the sum of $3$ odd positive composite integers.

The first few odd positive composite integers are:

$9, 15, 21, 25, 27, 33, 35, 39, 45$

Their differences with $47$ are:

$38, 32, 26, 22, 20, 14, 12, 8, 2$

The integers above are in the set of integers not expressible as a sum of $2$ odd positive composite integers.

The full set of these can be found in Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers.

Thus $18 k + 20$ is not expressible as the sum of $2 k$ odd positive composite integers.

This proves the result.

$\blacksquare$


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 the Shippensburg University Mathematical Problem Solving Group.


Sources

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