# 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:

\(\displaystyle m + n\) | \(=\) | \(\displaystyle \map \sigma {m_2}\) | Definition of Amicable Pair | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \sigma {2^s} \map \sigma b\) | Sigma Function is Multiplicative | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {2^{s + 1} - 1} \map \sigma b\) | Sigma Function of Power of Prime | ||||||||||

\(\displaystyle \) | \(\equiv\) | \(\displaystyle \paren {\paren {-1}^{s + 1} - 1} \map \sigma b\) | \(\displaystyle \pmod 3\) | Congruence of Powers |

Since $m + n$ is not divisible by $3$, $s$ must be even.

Similarly, by Sigma 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:

\(\displaystyle \frac {\map \sigma {m_1} } {m_1}\) | \(\ge\) | \(\displaystyle \frac {\map \sigma {12} } {12}\) | equality occurs if and only if $m_1 = 12$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac 7 3\) | |||||||||||

\(\displaystyle \frac {\map \sigma {m_2} } {m_2}\) | \(>\) | \(\displaystyle \frac {\map \sigma 4} 4\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac 7 4\) |

But:

\(\displaystyle 1\) | \(=\) | \(\displaystyle \frac {m_1 + m_2} {m_1 + m_2}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {m_1} {\map \sigma {m_1} } + \frac {m_2} {\map \sigma {m_2} }\) | Definition of Amicable Pair | ||||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle \frac 3 7 + \frac 4 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1\) |

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): $220$