Numbers in Even-Even Amicable Pair are not Divisible by 3
Theorem
Let $\tuple {m_1, m_2}$ be an amicable pair such that both $m_1$ and $m_2$ are even.
Then neither $m_1$ nor $m_2$ is divisible by $3$.
Proof
An amicable pair must be formed from a smaller abundant number and a larger deficient number.
Suppose both $m_1, m_2$ are divisible by $3$.
Since both are even, they must also be divisible by $6$.
However $6$ is a perfect number.
By Multiple of Perfect Number is Abundant, neither can be deficient.
So $m_1, m_2$ cannot form an amicable pair.
Therefore at most one of them is divisible by $3$.
Without loss of generality suppose $m_1$ is divisible by $3$.
Write:
- $m_1 = 2^r 3^t a, m_2 = 2^s b$
where $a, b$ are not divisible by $2$ or $3$.
Then:
\(\ds m + n\) | \(=\) | \(\ds \map {\sigma_1} {m_2}\) | Definition of Amicable Pair | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {2^s} \map {\sigma_1} b\) | Divisor Sum Function is Multiplicative | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2^{s + 1} - 1} \map {\sigma_1} b\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(\equiv\) | \(\ds \paren {\paren {-1}^{s + 1} - 1} \map {\sigma_1} b\) | \(\ds \pmod 3\) | Congruence of Powers |
Since $m + n$ is not divisible by $3$, $s$ must be even.
Similarly, by Divisor Sum Function is Multiplicative, $t$ must also be even.
In particular, both $s, t$ are at least $2$.
Now write:
- $m_1 = 2^2 \cdot 3 k, m_2 = 2^2 \cdot l$
where $k, l$ are some integers.
By Multiple of Perfect Number is Abundant, $m_1$ is abundant number.
Therefore $m_2 > m_1$.
This leads to $l > 3 k \ge 3$.
By Abundancy Index of Product is greater than Abundancy Index of Proper Factors:
\(\ds \frac {\map {\sigma_1} {m_1} } {m_1}\) | \(\ge\) | \(\ds \frac {\map {\sigma_1} {12} } {12}\) | equality occurs if and only if $m_1 = 12$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 7 3\) | ||||||||||||
\(\ds \frac {\map {\sigma_1} {m_2} } {m_2}\) | \(>\) | \(\ds \frac {\map {\sigma_1} 4} 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 7 4\) |
But:
\(\ds 1\) | \(=\) | \(\ds \frac {m_1 + m_2} {m_1 + m_2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {m_1} {\map {\sigma_1} {m_1} } + \frac {m_2} {\map {\sigma_1} {m_2} }\) | Definition of Amicable Pair | |||||||||||
\(\ds \) | \(<\) | \(\ds \frac 3 7 + \frac 4 7\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
which is a contradiction.
Therefore neither $m_1$ nor $m_2$ is divisible by $3$.
$\blacksquare$
Sources
- 1969: Elvin J. Lee: On Divisibility by Nine of the Sums of Even Amicable Pairs (Math. Comp. Vol. 23, no. 107: pp. 545 – 548)
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $220$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $220$