Even Integers not Sum of Two Abundant Numbers
Theorem
The even integers which are not the sum of $2$ abundant numbers are:
- All even integers less than $24$;
- $26, 28, 34, 46$
Proof
From Sequence of Abundant Numbers, the first few abundant numbers are:
- $12, 18, 20, 24, 30, 36, 40, 42, 48$
Immediately we see that any number less than $2 \times 12 = 24$ cannot be expressed as a sum of $2$ abundant numbers.
Next sum of $2$ abundant numbers is $12 + 18 = 30$, so $26$ and $28$ are not sums of $2$ abundant numbers.
| \(\ds 34 - 12\) | \(=\) | \(\ds 22\) | ||||||||||||
| \(\ds 34 - 18\) | \(<\) | \(\ds \frac {34} 2\) | ||||||||||||
| \(\ds 46 - 12\) | \(=\) | \(\ds 34\) | ||||||||||||
| \(\ds 46 - 18\) | \(=\) | \(\ds 28\) | ||||||||||||
| \(\ds 46 - 20\) | \(=\) | \(\ds 26\) | ||||||||||||
| \(\ds 46 - 24\) | \(<\) | \(\ds \frac {46} 2\) |
Since none of the differences above are abundant numbers, $34$ and $46$ are not sums of $2$ abundant numbers.
We demonstrate that $32$ and all even numbers from $36$ to $66$ except $46$ are sums of $2$ abundant numbers:
| \(\ds 32\) | \(=\) | \(\ds 12 + 20\) | ||||||||||||
| \(\ds 38\) | \(=\) | \(\ds 18 + 20\) | ||||||||||||
| \(\ds 40\) | \(=\) | \(\ds 20 + 20\) | ||||||||||||
| \(\ds 44\) | \(=\) | \(\ds 20 + 24\) | ||||||||||||
| \(\ds 50\) | \(=\) | \(\ds 20 + 30\) | ||||||||||||
| \(\ds 52\) | \(=\) | \(\ds 12 + 40\) | ||||||||||||
| \(\ds 56\) | \(=\) | \(\ds 20 + 36\) | ||||||||||||
| \(\ds 58\) | \(=\) | \(\ds 18 + 40\) | ||||||||||||
| \(\ds 62\) | \(=\) | \(\ds 20 + 42\) | ||||||||||||
| \(\ds 64\) | \(=\) | \(\ds 24 + 40\) |
The numbers $36, 42, 48, 54, 60$ and $66$ are multiples of $6$.
By Multiple of Perfect Number is Abundant, any multiple of $6$ greater than $6$ is abundant.
Hence these numbers can be expressed as:
- $12 + \paren {n - 12}$
which are sums of $2$ multiples of $6$ greater than $6$.
Now that we show that all even numbers greater than $66$ are sums of $2$ abundant numbers.
By Multiple of Abundant Number is Abundant, any multiple of $20$ is abundant.
By Largest Number not Expressible as Sum of Multiples of Coprime Integers/Generalization, any even number greater than:
- $\dfrac {6 \times 20} {\gcd \set {6, 20}} - 6 - 20 = 34$
is a sum of (possibly zero) multiples of $6$ and $20$.
Hence any even number greater than:
- $34 + 6 \times 2 + 20 = 66$
is a sum of a multiple of $6$ greater than $6$ and a non-zero multiple of $20$, which by above is a sum of $2$ abundant numbers.
This shows that the list above is complete.
$\blacksquare$
Also see
Sources
- 1964: Thomas R. Parkin and Leon J. Lander: Abundant Numbers
- April 1965: Review of Abundant Numbers (Math. Comp. Vol. 19, no. 90: p. 334)